Calculus Examples

$lim_{t \to 3} \frac{3 - t}{\sqrt{t + 1} - \sqrt{5 t - 11}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{t \to 3} 3 - t}{lim_{t \to 3} \sqrt{t + 1} - \sqrt{5 t - 11}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{lim_{t \to 3} 3 - lim_{t \to 3} t}{lim_{t \to 3} \sqrt{t + 1} - \sqrt{5 t - 11}}$

Step 1.1.2.2

Evaluate the limit of $3$ which is constant as $t$ approaches $3$ .

$\frac{3 - lim_{t \to 3} t}{lim_{t \to 3} \sqrt{t + 1} - \sqrt{5 t - 11}}$

Step 1.1.2.3

Simplify the expression.

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Step 1.1.2.3.1

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

$\frac{3 - 3}{lim_{t \to 3} \sqrt{t + 1} - \sqrt{5 t - 11}}$

Step 1.1.2.3.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{t \to 3} \sqrt{t + 1} - \sqrt{5 t - 11}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{0}{lim_{t \to 3} \sqrt{t + 1} - lim_{t \to 3} \sqrt{5 t - 11}}$

Step 1.1.3.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{t \to 3} t + 1} - lim_{t \to 3} \sqrt{5 t - 11}}$

Step 1.1.3.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{0}{\sqrt{lim_{t \to 3} t + lim_{t \to 3} 1} - lim_{t \to 3} \sqrt{5 t - 11}}$

Step 1.1.3.4

Evaluate the limit of $1$ which is constant as $t$ approaches $3$ .

$\frac{0}{\sqrt{lim_{t \to 3} t + 1} - lim_{t \to 3} \sqrt{5 t - 11}}$

Step 1.1.3.5

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{t \to 3} t + 1} - \sqrt{lim_{t \to 3} 5 t - 11}}$

Step 1.1.3.6

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{0}{\sqrt{lim_{t \to 3} t + 1} - \sqrt{lim_{t \to 3} 5 t - lim_{t \to 3} 11}}$

Step 1.1.3.7

Move the term $5$ outside of the limit because it is constant with respect to $t$ .

$\frac{0}{\sqrt{lim_{t \to 3} t + 1} - \sqrt{5 lim_{t \to 3} t - lim_{t \to 3} 11}}$

Step 1.1.3.8

Evaluate the limit of $11$ which is constant as $t$ approaches $3$ .

$\frac{0}{\sqrt{lim_{t \to 3} t + 1} - \sqrt{5 lim_{t \to 3} t - 1 \cdot 11}}$

Step 1.1.3.9

Evaluate the limits by plugging in $3$ for all occurrences of $t$ .

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Step 1.1.3.9.1

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

$\frac{0}{\sqrt{3 + 1} - \sqrt{5 lim_{t \to 3} t - 1 \cdot 11}}$

Step 1.1.3.9.2

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

$\frac{0}{\sqrt{3 + 1} - \sqrt{5 \cdot 3 - 1 \cdot 11}}$

Step 1.1.3.10

Simplify the answer.

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Step 1.1.3.10.1

Simplify each term.

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Step 1.1.3.10.1.1

Add $3$ and $1$ .

$\frac{0}{\sqrt{4} - \sqrt{5 \cdot 3 - 1 \cdot 11}}$

Step 1.1.3.10.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{0}{\sqrt{2^{2}} - \sqrt{5 \cdot 3 - 1 \cdot 11}}$

Step 1.1.3.10.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{2 - \sqrt{5 \cdot 3 - 1 \cdot 11}}$

Step 1.1.3.10.1.4

Multiply $5$ by $3$ .

$\frac{0}{2 - \sqrt{15 - 1 \cdot 11}}$

Step 1.1.3.10.1.5

Multiply $- 1$ by $11$ .

$\frac{0}{2 - \sqrt{15 - 11}}$

Step 1.1.3.10.1.6

Subtract $11$ from $15$ .

$\frac{0}{2 - \sqrt{4}}$

Step 1.1.3.10.1.7

Rewrite $4$ as $2^{2}$ .

$\frac{0}{2 - \sqrt{2^{2}}}$

Step 1.1.3.10.1.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0}{2 - 1 \cdot 2}$

Step 1.1.3.10.1.9

Multiply $- 1$ by $2$ .

$\frac{0}{2 - 2}$

Step 1.1.3.10.2

Subtract $2$ from $2$ .

$\frac{0}{0}$

Step 1.1.3.10.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.11

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{t \to 3} \frac{3 - t}{\sqrt{t + 1} - \sqrt{5 t - 11}} = lim_{t \to 3} \frac{\frac{d}{d t} [3 - t]}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{t \to 3} \frac{\frac{d}{d t} [3 - t]}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.2

By the Sum Rule, the derivative of $3 - t$ with respect to $t$ is $\frac{d}{d t} [3] + \frac{d}{d t} [- t]$ .

$lim_{t \to 3} \frac{\frac{d}{d t} [3] + \frac{d}{d t} [- t]}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.3

Since $3$ is constant with respect to $t$ , the derivative of $3$ with respect to $t$ is $0$ .

$lim_{t \to 3} \frac{0 + \frac{d}{d t} [- t]}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.4

Evaluate $\frac{d}{d t} [- t]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $t$ , the derivative of $- t$ with respect to $t$ is $- \frac{d}{d t} [t]$ .

$lim_{t \to 3} \frac{0 - \frac{d}{d t} [t]}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 3} \frac{0 - 1 \cdot 1}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.4.3

Multiply $- 1$ by $1$ .

$lim_{t \to 3} \frac{0 - 1}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.5

Subtract $1$ from $0$ .

$lim_{t \to 3} \frac{- 1}{\frac{d}{d t} [\sqrt{t + 1} - \sqrt{5 t - 11}]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{t + 1} - \sqrt{5 t - 11}$ with respect to $t$ is $\frac{d}{d t} [\sqrt{t + 1}] + \frac{d}{d t} [- \sqrt{5 t - 11}]$ .

$lim_{t \to 3} \frac{- 1}{\frac{d}{d t} [\sqrt{t + 1}] + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7

Evaluate $\frac{d}{d t} [\sqrt{t + 1}]$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t + 1}$ as ${(t + 1)}^{\frac{1}{2}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{d}{d t} [{(t + 1)}^{\frac{1}{2}}] + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = t + 1$ .

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Step 1.3.7.2.1

To apply the Chain Rule, set $u_{1}$ as $t + 1$ .

$lim_{t \to 3} \frac{- 1}{\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d t} [t + 1] + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 1] + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.2.3

Replace all occurrences of $u_{1}$ with $t + 1$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1}{2} - 1} \frac{d}{d t} [t + 1] + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.3

By the Sum Rule, the derivative of $t + 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [1]$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1}{2} - 1} (\frac{d}{d t} [t] + \frac{d}{d t} [1]) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1}{2} - 1} (1 + \frac{d}{d t} [1]) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.5

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1}{2} - 1} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.7

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.8

Combine the numerators over the common denominator.

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1 - 1 \cdot 2}{2}} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.9

Simplify the numerator.

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Step 1.3.7.9.1

Multiply $- 1$ by $2$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{1 - 2}{2}} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.9.2

Subtract $2$ from $1$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{\frac{- 1}{2}} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.10

Move the negative in front of the fraction.

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{- \frac{1}{2}} (1 + 0) + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.11

Add $1$ and $0$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2} {(t + 1)}^{- \frac{1}{2}} \cdot 1 + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.12

Combine $\frac{1}{2}$ and ${(t + 1)}^{- \frac{1}{2}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{{(t + 1)}^{- \frac{1}{2}}}{2} \cdot 1 + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.13

Multiply $\frac{{(t + 1)}^{- \frac{1}{2}}}{2}$ by $1$ .

$lim_{t \to 3} \frac{- 1}{\frac{{(t + 1)}^{- \frac{1}{2}}}{2} + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.7.14

Move ${(t + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} + \frac{d}{d t} [- \sqrt{5 t - 11}]}$

Step 1.3.8

Evaluate $\frac{d}{d t} [- \sqrt{5 t - 11}]$ .

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Step 1.3.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 t - 11}$ as ${(5 t - 11)}^{\frac{1}{2}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} + \frac{d}{d t} [- {(5 t - 11)}^{\frac{1}{2}}]}$

Step 1.3.8.2

Since $- 1$ is constant with respect to $t$ , the derivative of $- {(5 t - 11)}^{\frac{1}{2}}$ with respect to $t$ is $- \frac{d}{d t} [{(5 t - 11)}^{\frac{1}{2}}]$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - \frac{d}{d t} [{(5 t - 11)}^{\frac{1}{2}}]}$

Step 1.3.8.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{\frac{1}{2}}$ and $g (t) = 5 t - 11$ .

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Step 1.3.8.3.1

To apply the Chain Rule, set $u_{2}$ as $5 t - 11$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d t} [5 t - 11])}$

Step 1.3.8.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{2}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d t} [5 t - 11])}$

Step 1.3.8.3.3

Replace all occurrences of $u_{2}$ with $5 t - 11$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} - 1} \frac{d}{d t} [5 t - 11])}$

Step 1.3.8.4

By the Sum Rule, the derivative of $5 t - 11$ with respect to $t$ is $\frac{d}{d t} [5 t] + \frac{d}{d t} [- 11]$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} - 1} (\frac{d}{d t} [5 t] + \frac{d}{d t} [- 11]))}$

Step 1.3.8.5

Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} - 1} (5 \frac{d}{d t} [t] + \frac{d}{d t} [- 11]))}$

Step 1.3.8.6

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} - 1} (5 \cdot 1 + \frac{d}{d t} [- 11]))}$

Step 1.3.8.7

Since $- 11$ is constant with respect to $t$ , the derivative of $- 11$ with respect to $t$ is $0$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} - 1} (5 \cdot 1 + 0))}$

Step 1.3.8.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (5 \cdot 1 + 0))}$

Step 1.3.8.9

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (5 \cdot 1 + 0))}$

Step 1.3.8.10

Combine the numerators over the common denominator.

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1 - 1 \cdot 2}{2}} (5 \cdot 1 + 0))}$

Step 1.3.8.11

Simplify the numerator.

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Step 1.3.8.11.1

Multiply $- 1$ by $2$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{1 - 2}{2}} (5 \cdot 1 + 0))}$

Step 1.3.8.11.2

Subtract $2$ from $1$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{\frac{- 1}{2}} (5 \cdot 1 + 0))}$

Step 1.3.8.12

Move the negative in front of the fraction.

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{- \frac{1}{2}} (5 \cdot 1 + 0))}$

Step 1.3.8.13

Multiply $5$ by $1$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{- \frac{1}{2}} (5 + 0))}$

Step 1.3.8.14

Add $5$ and $0$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{1}{2} {(5 t - 11)}^{- \frac{1}{2}} \cdot 5)}$

Step 1.3.8.15

Combine $\frac{1}{2}$ and ${(5 t - 11)}^{- \frac{1}{2}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - (\frac{{(5 t - 11)}^{- \frac{1}{2}}}{2} \cdot 5)}$

Step 1.3.8.16

Combine $\frac{{(5 t - 11)}^{- \frac{1}{2}}}{2}$ and $5$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - \frac{{(5 t - 11)}^{- \frac{1}{2}} \cdot 5}{2}}$

Step 1.3.8.17

Move $5$ to the left of ${(5 t - 11)}^{- \frac{1}{2}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - \frac{5 \cdot {(5 t - 11)}^{- \frac{1}{2}}}{2}}$

Step 1.3.8.18

Move ${(5 t - 11)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 {(t + 1)}^{\frac{1}{2}}} - \frac{5}{2 {(5 t - 11)}^{\frac{1}{2}}}}$

Step 1.4

Convert fractional exponents to radicals.

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Step 1.4.1

Rewrite ${(t + 1)}^{\frac{1}{2}}$ as $\sqrt{t + 1}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 \sqrt{t + 1}} - \frac{5}{2 {(5 t - 11)}^{\frac{1}{2}}}}$

Step 1.4.2

Rewrite ${(5 t - 11)}^{\frac{1}{2}}$ as $\sqrt{5 t - 11}$ .

$lim_{t \to 3} \frac{- 1}{\frac{1}{2 \sqrt{t + 1}} - \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $3$ .

$\frac{lim_{t \to 3} - 1}{lim_{t \to 3} \frac{1}{2 \sqrt{t + 1}} - \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.2

Evaluate the limit of $- 1$ which is constant as $t$ approaches $3$ .

$\frac{- 1}{lim_{t \to 3} \frac{1}{2 \sqrt{t + 1}} - \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{- 1}{lim_{t \to 3} \frac{1}{2 \sqrt{t + 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $t$ .

$\frac{- 1}{\frac{1}{2} lim_{t \to 3} \frac{1}{\sqrt{t + 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.5

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{lim_{t \to 3} 1}{lim_{t \to 3} \sqrt{t + 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.6

Evaluate the limit of $1$ which is constant as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{lim_{t \to 3} \sqrt{t + 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.7

Move the limit under the radical sign.

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.8

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + lim_{t \to 3} 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.9

Evaluate the limit of $1$ which is constant as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - lim_{t \to 3} \frac{5}{2 \sqrt{5 t - 11}}}$

Step 2.10

Move the term $\frac{5}{2}$ outside of the limit because it is constant with respect to $t$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} lim_{t \to 3} \frac{1}{\sqrt{5 t - 11}}}$

Step 2.11

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} \cdot \frac{lim_{t \to 3} 1}{lim_{t \to 3} \sqrt{5 t - 11}}}$

Step 2.12

Evaluate the limit of $1$ which is constant as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} \cdot \frac{1}{lim_{t \to 3} \sqrt{5 t - 11}}}$

Step 2.13

Move the limit under the radical sign.

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} 5 t - 11}}}$

Step 2.14

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} 5 t - lim_{t \to 3} 11}}}$

Step 2.15

Move the term $5$ outside of the limit because it is constant with respect to $t$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 lim_{t \to 3} t - lim_{t \to 3} 11}}}$

Step 2.16

Evaluate the limit of $11$ which is constant as $t$ approaches $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{t \to 3} t + 1}} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 lim_{t \to 3} t - 1 \cdot 11}}}$

Step 3

Evaluate the limits by plugging in $3$ for all occurrences of $t$ .

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Step 3.1

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{3 + 1}} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 lim_{t \to 3} t - 1 \cdot 11}}}$

Step 3.2

Evaluate the limit of $t$ by plugging in $3$ for $t$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{3 + 1}} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 \cdot 3 - 1 \cdot 11}}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Add $3$ and $1$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{4}} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 \cdot 3 - 1 \cdot 11}}}$

Step 4.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{\sqrt{2^{2}}} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 \cdot 3 - 1 \cdot 11}}}$

Step 4.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{2} - \frac{5}{2} \cdot \frac{1}{\sqrt{5 \cdot 3 - 1 \cdot 11}}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $5$ by $3$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{2} - \frac{5}{2} \cdot \frac{1}{\sqrt{15 - 1 \cdot 11}}}$

Step 4.2.2

Multiply $- 1$ by $11$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{2} - \frac{5}{2} \cdot \frac{1}{\sqrt{15 - 11}}}$

Step 4.2.3

Subtract $11$ from $15$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{2} - \frac{5}{2} \cdot \frac{1}{\sqrt{4}}}$

Step 4.2.4

Rewrite $4$ as $2^{2}$ .

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{2} - \frac{5}{2} \cdot \frac{1}{\sqrt{2^{2}}}}$

Step 4.2.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- 1}{\frac{1}{2} \cdot \frac{1}{2} - \frac{5}{2} \cdot \frac{1}{2}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 4.3.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{- 1}{\frac{1}{2 \cdot 2} - \frac{5}{2} \cdot \frac{1}{2}}$

Step 4.3.1.2

Multiply $2$ by $2$ .

$\frac{- 1}{\frac{1}{4} - \frac{5}{2} \cdot \frac{1}{2}}$

Step 4.3.2

Multiply $- \frac{5}{2} \cdot \frac{1}{2}$ .

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Step 4.3.2.1

Multiply $\frac{1}{2}$ by $\frac{5}{2}$ .

$\frac{- 1}{\frac{1}{4} - \frac{5}{2 \cdot 2}}$

Step 4.3.2.2

Multiply $2$ by $2$ .

$\frac{- 1}{\frac{1}{4} - \frac{5}{4}}$

Step 4.3.3

Combine the numerators over the common denominator.

$\frac{- 1}{\frac{1 - 5}{4}}$

Step 4.3.4

Subtract $5$ from $1$ .

$\frac{- 1}{\frac{- 4}{4}}$

Step 4.3.5

Divide $- 4$ by $4$ .

$\frac{- 1}{- 1}$

Step 4.4

Divide $- 1$ by $- 1$ .

$1$