Calculus Examples

$lim_{t \to 3} \frac{t^{3} + 19 t^{2} - 68 t + 6}{t^{3} - 9 t}$

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.

$\frac{lim_{t \to 3} t^{3} + 19 t^{2} - 68 t + 6}{lim_{t \to 3} t^{3} - 9 t}$

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 $3$ .

$\frac{lim_{t \to 3} t^{3} + lim_{t \to 3} 19 t^{2} - lim_{t \to 3} 68 t + lim_{t \to 3} 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.2

Move the exponent $3$ from $t^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{t \to 3} t)}^{3} + lim_{t \to 3} 19 t^{2} - lim_{t \to 3} 68 t + lim_{t \to 3} 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.3

Move the term $19$ outside of the limit because it is constant with respect to $t$ .

$\frac{{(lim_{t \to 3} t)}^{3} + 19 lim_{t \to 3} t^{2} - lim_{t \to 3} 68 t + lim_{t \to 3} 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.4

Move the exponent $2$ from $t^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{t \to 3} t)}^{3} + 19 {(lim_{t \to 3} t)}^{2} - lim_{t \to 3} 68 t + lim_{t \to 3} 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.5

Move the term $68$ outside of the limit because it is constant with respect to $t$ .

$\frac{{(lim_{t \to 3} t)}^{3} + 19 {(lim_{t \to 3} t)}^{2} - 68 lim_{t \to 3} t + lim_{t \to 3} 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.6

Evaluate the limit of $6$ which is constant as $t$ approaches $3$ .

$\frac{{(lim_{t \to 3} t)}^{3} + 19 {(lim_{t \to 3} t)}^{2} - 68 lim_{t \to 3} t + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.7

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

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Step 1.1.2.7.1

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

$\frac{3^{3} + 19 {(lim_{t \to 3} t)}^{2} - 68 lim_{t \to 3} t + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.7.2

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

$\frac{3^{3} + 19 \cdot 3^{2} - 68 lim_{t \to 3} t + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.7.3

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

$\frac{3^{3} + 19 \cdot 3^{2} - 68 \cdot 3 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Simplify each term.

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Step 1.1.2.8.1.1

Raise $3$ to the power of $3$ .

$\frac{27 + 19 \cdot 3^{2} - 68 \cdot 3 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8.1.2

Raise $3$ to the power of $2$ .

$\frac{27 + 19 \cdot 9 - 68 \cdot 3 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8.1.3

Multiply $19$ by $9$ .

$\frac{27 + 171 - 68 \cdot 3 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8.1.4

Multiply $- 68$ by $3$ .

$\frac{27 + 171 - 204 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8.2

Add $27$ and $171$ .

$\frac{198 - 204 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8.3

Subtract $204$ from $198$ .

$\frac{- 6 + 6}{lim_{t \to 3} t^{3} - 9 t}$

Step 1.1.2.8.4

Add $- 6$ and $6$ .

$\frac{0}{lim_{t \to 3} t^{3} - 9 t}$

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 $3$ .

$\frac{0}{lim_{t \to 3} t^{3} - lim_{t \to 3} 9 t}$

Step 1.1.3.2

Move the exponent $3$ from $t^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{t \to 3} t)}^{3} - lim_{t \to 3} 9 t}$

Step 1.1.3.3

Move the term $9$ outside of the limit because it is constant with respect to $t$ .

$\frac{0}{{(lim_{t \to 3} t)}^{3} - 9 lim_{t \to 3} t}$

Step 1.1.3.4

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

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Step 1.1.3.4.1

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

$\frac{0}{3^{3} - 9 lim_{t \to 3} t}$

Step 1.1.3.4.2

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

$\frac{0}{3^{3} - 9 \cdot 3}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

Raise $3$ to the power of $3$ .

$\frac{0}{27 - 9 \cdot 3}$

Step 1.1.3.5.1.2

Multiply $- 9$ by $3$ .

$\frac{0}{27 - 27}$

Step 1.1.3.5.2

Subtract $27$ from $27$ .

$\frac{0}{0}$

Step 1.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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.

$lim_{t \to 3} \frac{t^{3} + 19 t^{2} - 68 t + 6}{t^{3} - 9 t} = lim_{t \to 3} \frac{\frac{d}{d t} [t^{3} + 19 t^{2} - 68 t + 6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{t \to 3} \frac{\frac{d}{d t} [t^{3} + 19 t^{2} - 68 t + 6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.2

By the Sum Rule, the derivative of $t^{3} + 19 t^{2} - 68 t + 6$ with respect to $t$ is $\frac{d}{d t} [t^{3}] + \frac{d}{d t} [19 t^{2}] + \frac{d}{d t} [- 68 t] + \frac{d}{d t} [6]$ .

$lim_{t \to 3} \frac{\frac{d}{d t} [t^{3}] + \frac{d}{d t} [19 t^{2}] + \frac{d}{d t} [- 68 t] + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$lim_{t \to 3} \frac{3 t^{2} + \frac{d}{d t} [19 t^{2}] + \frac{d}{d t} [- 68 t] + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.4

Evaluate $\frac{d}{d t} [19 t^{2}]$ .

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Step 1.3.4.1

Since $19$ is constant with respect to $t$ , the derivative of $19 t^{2}$ with respect to $t$ is $19 \frac{d}{d t} [t^{2}]$ .

$lim_{t \to 3} \frac{3 t^{2} + 19 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 68 t] + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

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 $n = 2$ .

$lim_{t \to 3} \frac{3 t^{2} + 19 (2 t) + \frac{d}{d t} [- 68 t] + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.4.3

Multiply $2$ by $19$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t + \frac{d}{d t} [- 68 t] + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.5

Evaluate $\frac{d}{d t} [- 68 t]$ .

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Step 1.3.5.1

Since $- 68$ is constant with respect to $t$ , the derivative of $- 68 t$ with respect to $t$ is $- 68 \frac{d}{d t} [t]$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68 \frac{d}{d t} [t] + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.5.2

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

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68 \cdot 1 + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.5.3

Multiply $- 68$ by $1$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68 + \frac{d}{d t} [6]}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.6

Since $6$ is constant with respect to $t$ , the derivative of $6$ with respect to $t$ is $0$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68 + 0}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.7

Add $3 t^{2} + 38 t - 68$ and $0$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68}{\frac{d}{d t} [t^{3} - 9 t]}$

Step 1.3.8

By the Sum Rule, the derivative of $t^{3} - 9 t$ with respect to $t$ is $\frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 9 t]$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68}{\frac{d}{d t} [t^{3}] + \frac{d}{d t} [- 9 t]}$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68}{3 t^{2} + \frac{d}{d t} [- 9 t]}$

Step 1.3.10

Evaluate $\frac{d}{d t} [- 9 t]$ .

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Step 1.3.10.1

Since $- 9$ is constant with respect to $t$ , the derivative of $- 9 t$ with respect to $t$ is $- 9 \frac{d}{d t} [t]$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68}{3 t^{2} - 9 \frac{d}{d t} [t]}$

Step 1.3.10.2

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

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68}{3 t^{2} - 9 \cdot 1}$

Step 1.3.10.3

Multiply $- 9$ by $1$ .

$lim_{t \to 3} \frac{3 t^{2} + 38 t - 68}{3 t^{2} - 9}$

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 $3$ .

$\frac{lim_{t \to 3} 3 t^{2} + 38 t - 68}{lim_{t \to 3} 3 t^{2} - 9}$

Step 2.2

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

$\frac{lim_{t \to 3} 3 t^{2} + lim_{t \to 3} 38 t - lim_{t \to 3} 68}{lim_{t \to 3} 3 t^{2} - 9}$

Step 2.3

Move the term $3$ outside of the limit because it is constant with respect to $t$ .

$\frac{3 lim_{t \to 3} t^{2} + lim_{t \to 3} 38 t - lim_{t \to 3} 68}{lim_{t \to 3} 3 t^{2} - 9}$

Step 2.4

Move the exponent $2$ from $t^{2}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{t \to 3} t)}^{2} + lim_{t \to 3} 38 t - lim_{t \to 3} 68}{lim_{t \to 3} 3 t^{2} - 9}$

Step 2.5

Move the term $38$ outside of the limit because it is constant with respect to $t$ .

$\frac{3 {(lim_{t \to 3} t)}^{2} + 38 lim_{t \to 3} t - lim_{t \to 3} 68}{lim_{t \to 3} 3 t^{2} - 9}$

Step 2.6

Evaluate the limit of $68$ which is constant as $t$ approaches $3$ .

$\frac{3 {(lim_{t \to 3} t)}^{2} + 38 lim_{t \to 3} t - 1 \cdot 68}{lim_{t \to 3} 3 t^{2} - 9}$

Step 2.7

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

$\frac{3 {(lim_{t \to 3} t)}^{2} + 38 lim_{t \to 3} t - 1 \cdot 68}{lim_{t \to 3} 3 t^{2} - lim_{t \to 3} 9}$

Step 2.8

Move the term $3$ outside of the limit because it is constant with respect to $t$ .

$\frac{3 {(lim_{t \to 3} t)}^{2} + 38 lim_{t \to 3} t - 1 \cdot 68}{3 lim_{t \to 3} t^{2} - lim_{t \to 3} 9}$

Step 2.9

Move the exponent $2$ from $t^{2}$ outside the limit using the Limits Power Rule.

$\frac{3 {(lim_{t \to 3} t)}^{2} + 38 lim_{t \to 3} t - 1 \cdot 68}{3 {(lim_{t \to 3} t)}^{2} - lim_{t \to 3} 9}$

Step 2.10

Evaluate the limit of $9$ which is constant as $t$ approaches $3$ .

$\frac{3 {(lim_{t \to 3} t)}^{2} + 38 lim_{t \to 3} t - 1 \cdot 68}{3 {(lim_{t \to 3} t)}^{2} - 1 \cdot 9}$

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 $t$ .

$\frac{3 \cdot 3^{2} + 38 lim_{t \to 3} t - 1 \cdot 68}{3 {(lim_{t \to 3} t)}^{2} - 1 \cdot 9}$

Step 3.2

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

$\frac{3 \cdot 3^{2} + 38 \cdot 3 - 1 \cdot 68}{3 {(lim_{t \to 3} t)}^{2} - 1 \cdot 9}$

Step 3.3

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

$\frac{3 \cdot 3^{2} + 38 \cdot 3 - 1 \cdot 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 4.1.1.1

Multiply $3$ by $3^{2}$ .

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Step 4.1.1.1.1

Raise $3$ to the power of $1$ .

$\frac{3^{1} \cdot 3^{2} + 38 \cdot 3 - 1 \cdot 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3^{1 + 2} + 38 \cdot 3 - 1 \cdot 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.1.2

Add $1$ and $2$ .

$\frac{3^{3} + 38 \cdot 3 - 1 \cdot 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.2

Raise $3$ to the power of $3$ .

$\frac{27 + 38 \cdot 3 - 1 \cdot 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.3

Multiply $38$ by $3$ .

$\frac{27 + 114 - 1 \cdot 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.4

Multiply $- 1$ by $68$ .

$\frac{27 + 114 - 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.5

Add $27$ and $114$ .

$\frac{141 - 68}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.1.6

Subtract $68$ from $141$ .

$\frac{73}{3 \cdot 3^{2} - 1 \cdot 9}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 4.2.1.1

Multiply $3$ by $3^{2}$ .

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Step 4.2.1.1.1

Raise $3$ to the power of $1$ .

$\frac{73}{3^{1} \cdot 3^{2} - 1 \cdot 9}$

Step 4.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{73}{3^{1 + 2} - 1 \cdot 9}$

Step 4.2.1.2

Add $1$ and $2$ .

$\frac{73}{3^{3} - 1 \cdot 9}$

Step 4.2.2

Raise $3$ to the power of $3$ .

$\frac{73}{27 - 1 \cdot 9}$

Step 4.2.3

Multiply $- 1$ by $9$ .

$\frac{73}{27 - 9}$

Step 4.2.4

Subtract $9$ from $27$ .

$\frac{73}{18}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{73}{18}$

Decimal Form:

$4.0\overline{5}$