Calculus Examples

$lim_{t \to 0} \frac{2}{t} - \frac{2}{t^{2} + t}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{2}{t}$ as a fraction with a common denominator, multiply by $\frac{t^{2} + t}{t^{2} + t}$ .

$lim_{t \to 0} \frac{2}{t} \cdot \frac{t^{2} + t}{t^{2} + t} - \frac{2}{t^{2} + t}$

Step 1.2

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

$lim_{t \to 0} \frac{2}{t} \cdot \frac{t^{2} + t}{t^{2} + t} - \frac{2}{t^{2} + t} \cdot \frac{t}{t}$

Step 1.3

Write each expression with a common denominator of $t (t^{2} + t)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2}{t}$ by $\frac{t^{2} + t}{t^{2} + t}$ .

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

Step 1.3.2

Multiply $\frac{2}{t^{2} + t}$ by $\frac{t}{t}$ .

$lim_{t \to 0} \frac{2 (t^{2} + t)}{t (t^{2} + t)} - \frac{2 t}{(t^{2} + t) t}$

Step 1.3.3

Reorder the factors of $(t^{2} + t) t$ .

$lim_{t \to 0} \frac{2 (t^{2} + t)}{t (t^{2} + t)} - \frac{2 t}{t (t^{2} + t)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{t \to 0} \frac{2 (t^{2} + t) - 2 t}{t (t^{2} + t)}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{t \to 0} 2 (t^{2} + t) - 2 t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$\frac{lim_{t \to 0} 2 (t^{2} + t) - lim_{t \to 0} 2 t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.2

Move the term $2$ outside of the limit because it is constant with respect to $t$ .

$\frac{2 lim_{t \to 0} t^{2} + t - lim_{t \to 0} 2 t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$\frac{2 (lim_{t \to 0} t^{2} + lim_{t \to 0} t) - lim_{t \to 0} 2 t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.4

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

$\frac{2 ({(lim_{t \to 0} t)}^{2} + lim_{t \to 0} t) - lim_{t \to 0} 2 t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.5

Move the term $2$ outside of the limit because it is constant with respect to $t$ .

$\frac{2 ({(lim_{t \to 0} t)}^{2} + lim_{t \to 0} t) - 2 lim_{t \to 0} t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.6

Evaluate the limits by plugging in $0$ for all occurrences of $t$ .

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Step 2.1.2.6.1

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{2 (0^{2} + lim_{t \to 0} t) - 2 lim_{t \to 0} t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.6.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{2 (0^{2} + 0) - 2 lim_{t \to 0} t}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.6.3

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

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

Step 2.1.2.7

Simplify the answer.

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Step 2.1.2.7.1

Simplify each term.

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Step 2.1.2.7.1.1

Raising $0$ to any positive power yields $0$ .

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

Step 2.1.2.7.1.2

Add $0$ and $0$ .

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

Step 2.1.2.7.1.3

Multiply $2$ by $0$ .

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

Step 2.1.2.7.1.4

Multiply $- 2$ by $0$ .

$\frac{0 + 0}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.2.7.2

Add $0$ and $0$ .

$\frac{0}{lim_{t \to 0} t (t^{2} + t)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Product of Limits Rule on the limit as $t$ approaches $0$ .

$\frac{0}{lim_{t \to 0} t \cdot lim_{t \to 0} t^{2} + t}$

Step 2.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$\frac{0}{lim_{t \to 0} t \cdot (lim_{t \to 0} t^{2} + lim_{t \to 0} t)}$

Step 2.1.3.3

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

$\frac{0}{lim_{t \to 0} t \cdot ({(lim_{t \to 0} t)}^{2} + lim_{t \to 0} t)}$

Step 2.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $t$ .

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Step 2.1.3.4.1

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0}{0 \cdot ({(lim_{t \to 0} t)}^{2} + lim_{t \to 0} t)}$

Step 2.1.3.4.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0}{0 \cdot (0^{2} + lim_{t \to 0} t)}$

Step 2.1.3.4.3

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0}{0 \cdot (0^{2} + 0)}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 \cdot (0 + 0)}$

Step 2.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.5.3

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{t \to 0} \frac{2 (t^{2} + t) - 2 t}{t (t^{2} + t)} = lim_{t \to 0} \frac{\frac{d}{d t} [2 (t^{2} + t) - 2 t]}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{t \to 0} \frac{\frac{d}{d t} [2 (t^{2} + t) - 2 t]}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.2

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

$lim_{t \to 0} \frac{\frac{d}{d t} [2 (t^{2} + t)] + \frac{d}{d t} [- 2 t]}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.3

Evaluate $\frac{d}{d t} [2 (t^{2} + t)]$ .

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Step 2.3.3.1

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

$lim_{t \to 0} \frac{2 \frac{d}{d t} [t^{2} + t] + \frac{d}{d t} [- 2 t]}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.3.2

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

$lim_{t \to 0} \frac{2 (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [t]) + \frac{d}{d t} [- 2 t]}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.3.3

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

$lim_{t \to 0} \frac{2 (2 t + \frac{d}{d t} [t]) + \frac{d}{d t} [- 2 t]}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.3.4

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

Multiply $- 2$ by $1$ .

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Apply the distributive property.

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

Step 2.3.5.2

Combine terms.

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Step 2.3.5.2.1

Multiply $2$ by $2$ .

$lim_{t \to 0} \frac{4 t + 2 \cdot 1 - 2}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.5.2.2

Multiply $2$ by $1$ .

$lim_{t \to 0} \frac{4 t + 2 - 2}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.5.2.3

Subtract $2$ from $2$ .

$lim_{t \to 0} \frac{4 t + 0}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.5.2.4

Add $4 t$ and $0$ .

$lim_{t \to 0} \frac{4 t}{\frac{d}{d t} [t (t^{2} + t)]}$

Step 2.3.6

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = t^{2} + t$ .

$lim_{t \to 0} \frac{4 t}{t \frac{d}{d t} [t^{2} + t] + (t^{2} + t) \frac{d}{d t} [t]}$

Step 2.3.7

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

$lim_{t \to 0} \frac{4 t}{t (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [t]) + (t^{2} + t) \frac{d}{d t} [t]}$

Step 2.3.8

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

$lim_{t \to 0} \frac{4 t}{t (2 t + \frac{d}{d t} [t]) + (t^{2} + t) \frac{d}{d t} [t]}$

Step 2.3.9

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

$lim_{t \to 0} \frac{4 t}{t (2 t + 1) + (t^{2} + t) \frac{d}{d t} [t]}$

Step 2.3.10

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

$lim_{t \to 0} \frac{4 t}{t (2 t + 1) + (t^{2} + t) \cdot 1}$

Step 2.3.11

Multiply $t^{2} + t$ by $1$ .

$lim_{t \to 0} \frac{4 t}{t (2 t + 1) + t^{2} + t}$

Step 2.3.12

Simplify.

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Step 2.3.12.1

Apply the distributive property.

$lim_{t \to 0} \frac{4 t}{t (2 t) + t \cdot 1 + t^{2} + t}$

Step 2.3.12.2

Combine terms.

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Step 2.3.12.2.1

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

$lim_{t \to 0} \frac{4 t}{2 (t^{1} t) + t \cdot 1 + t^{2} + t}$

Step 2.3.12.2.2

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

$lim_{t \to 0} \frac{4 t}{2 (t^{1} t^{1}) + t \cdot 1 + t^{2} + t}$

Step 2.3.12.2.3

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

$lim_{t \to 0} \frac{4 t}{2 t^{1 + 1} + t \cdot 1 + t^{2} + t}$

Step 2.3.12.2.4

Add $1$ and $1$ .

$lim_{t \to 0} \frac{4 t}{2 t^{2} + t \cdot 1 + t^{2} + t}$

Step 2.3.12.2.5

Multiply $t$ by $1$ .

$lim_{t \to 0} \frac{4 t}{2 t^{2} + t + t^{2} + t}$

Step 2.3.12.2.6

Add $2 t^{2}$ and $t^{2}$ .

$lim_{t \to 0} \frac{4 t}{3 t^{2} + t + t}$

Step 2.3.12.2.7

Add $t$ and $t$ .

$lim_{t \to 0} \frac{4 t}{3 t^{2} + 2 t}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $t$ .

$4 lim_{t \to 0} \frac{t}{3 t^{2} + 2 t}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

$4 \frac{lim_{t \to 0} t}{lim_{t \to 0} 3 t^{2} + 2 t}$

Step 4.1.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$4 \frac{0}{lim_{t \to 0} 3 t^{2} + 2 t}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$4 \frac{0}{lim_{t \to 0} 3 t^{2} + lim_{t \to 0} 2 t}$

Step 4.1.3.2

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

$4 \frac{0}{3 lim_{t \to 0} t^{2} + lim_{t \to 0} 2 t}$

Step 4.1.3.3

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

$4 \frac{0}{3 {(lim_{t \to 0} t)}^{2} + lim_{t \to 0} 2 t}$

Step 4.1.3.4

Move the term $2$ outside of the limit because it is constant with respect to $t$ .

$4 \frac{0}{3 {(lim_{t \to 0} t)}^{2} + 2 lim_{t \to 0} t}$

Step 4.1.3.5

Evaluate the limits by plugging in $0$ for all occurrences of $t$ .

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Step 4.1.3.5.1

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$4 \frac{0}{3 \cdot 0^{2} + 2 lim_{t \to 0} t}$

Step 4.1.3.5.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$4 \frac{0}{3 \cdot 0^{2} + 2 \cdot 0}$

Step 4.1.3.6

Simplify the answer.

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Step 4.1.3.6.1

Simplify each term.

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Step 4.1.3.6.1.1

Raising $0$ to any positive power yields $0$ .

$4 \frac{0}{3 \cdot 0 + 2 \cdot 0}$

Step 4.1.3.6.1.2

Multiply $3$ by $0$ .

$4 \frac{0}{0 + 2 \cdot 0}$

Step 4.1.3.6.1.3

Multiply $2$ by $0$ .

$4 \frac{0}{0 + 0}$

Step 4.1.3.6.2

Add $0$ and $0$ .

$4 (\frac{0}{0})$

Step 4.1.3.6.3

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

Undefined

$4 (\frac{0}{0})$

Step 4.1.3.7

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

Undefined

$4 (\frac{0}{0})$

Step 4.1.4

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

Undefined

$4 (\frac{0}{0})$

Step 4.2

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.

$lim_{t \to 0} \frac{t}{3 t^{2} + 2 t} = lim_{t \to 0} \frac{\frac{d}{d t} [t]}{\frac{d}{d t} [3 t^{2} + 2 t]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$4 lim_{t \to 0} \frac{\frac{d}{d t} [t]}{\frac{d}{d t} [3 t^{2} + 2 t]}$

Step 4.3.2

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

$4 lim_{t \to 0} \frac{1}{\frac{d}{d t} [3 t^{2} + 2 t]}$

Step 4.3.3

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

$4 lim_{t \to 0} \frac{1}{\frac{d}{d t} [3 t^{2}] + \frac{d}{d t} [2 t]}$

Step 4.3.4

Evaluate $\frac{d}{d t} [3 t^{2}]$ .

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Step 4.3.4.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t^{2}$ with respect to $t$ is $3 \frac{d}{d t} [t^{2}]$ .

$4 lim_{t \to 0} \frac{1}{3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [2 t]}$

Step 4.3.4.2

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

$4 lim_{t \to 0} \frac{1}{3 \cdot 2 t + \frac{d}{d t} [2 t]}$

Step 4.3.4.3

Multiply $2$ by $3$ .

$4 lim_{t \to 0} \frac{1}{6 t + \frac{d}{d t} [2 t]}$

Step 4.3.5

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

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Step 4.3.5.1

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

$4 lim_{t \to 0} \frac{1}{6 t + 2 \frac{d}{d t} [t]}$

Step 4.3.5.2

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

$4 lim_{t \to 0} \frac{1}{6 t + 2 \cdot 1}$

Step 4.3.5.3

Multiply $2$ by $1$ .

$4 lim_{t \to 0} \frac{1}{6 t + 2}$

Step 5

Evaluate the limit.

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Step 5.1

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $0$ .

$4 \frac{lim_{t \to 0} 1}{lim_{t \to 0} 6 t + 2}$

Step 5.2

Evaluate the limit of $1$ which is constant as $t$ approaches $0$ .

$4 \frac{1}{lim_{t \to 0} 6 t + 2}$

Step 5.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$4 \frac{1}{lim_{t \to 0} 6 t + lim_{t \to 0} 2}$

Step 5.4

Move the term $6$ outside of the limit because it is constant with respect to $t$ .

$4 \frac{1}{6 lim_{t \to 0} t + lim_{t \to 0} 2}$

Step 5.5

Evaluate the limit of $2$ which is constant as $t$ approaches $0$ .

$4 \frac{1}{6 lim_{t \to 0} t + 2}$

Step 6

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$4 \frac{1}{6 \cdot 0 + 2}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Multiply $6$ by $0$ .

$4 \frac{1}{0 + 2}$

Step 7.1.2

Add $0$ and $2$ .

$4 (\frac{1}{2})$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{1}{2}$

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$2$