Calculus Examples

$lim_{t \to 0} \frac{\sin (t)}{e^{3 t} - 1}$

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.

$\frac{lim_{t \to 0} \sin (t)}{lim_{t \to 0} e^{3 t} - 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Move the limit inside the trig function because sine is continuous.

$\frac{\sin (lim_{t \to 0} t)}{lim_{t \to 0} e^{3 t} - 1}$

Step 1.1.2.2

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

$\frac{\sin (0)}{lim_{t \to 0} e^{3 t} - 1}$

Step 1.1.2.3

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{lim_{t \to 0} e^{3 t} - 1}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{t \to 0} e^{3 t} - lim_{t \to 0} 1}$

Step 1.1.3.1.2

Move the limit into the exponent.

$\frac{0}{e^{lim_{t \to 0} 3 t} - lim_{t \to 0} 1}$

Step 1.1.3.1.3

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

$\frac{0}{e^{3 lim_{t \to 0} t} - lim_{t \to 0} 1}$

Step 1.1.3.1.4

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

$\frac{0}{e^{3 lim_{t \to 0} t} - 1 \cdot 1}$

Step 1.1.3.2

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

$\frac{0}{e^{3 \cdot 0} - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $3$ by $0$ .

$\frac{0}{e^{0} - 1 \cdot 1}$

Step 1.1.3.3.1.2

Anything raised to $0$ is $1$ .

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

Step 1.1.3.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{t \to 0} \frac{\sin (t)}{e^{3 t} - 1} = lim_{t \to 0} \frac{\frac{d}{d t} [\sin (t)]}{\frac{d}{d t} [e^{3 t} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{t \to 0} \frac{\frac{d}{d t} [\sin (t)]}{\frac{d}{d t} [e^{3 t} - 1]}$

Step 1.3.2

The derivative of $\sin (t)$ with respect to $t$ is $\cos (t)$ .

$lim_{t \to 0} \frac{\cos (t)}{\frac{d}{d t} [e^{3 t} - 1]}$

Step 1.3.3

By the Sum Rule, the derivative of $e^{3 t} - 1$ with respect to $t$ is $\frac{d}{d t} [e^{3 t}] + \frac{d}{d t} [- 1]$ .

$lim_{t \to 0} \frac{\cos (t)}{\frac{d}{d t} [e^{3 t}] + \frac{d}{d t} [- 1]}$

Step 1.3.4

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

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Step 1.3.4.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = 3 t$ .

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Step 1.3.4.1.1

To apply the Chain Rule, set $u$ as $3 t$ .

$lim_{t \to 0} \frac{\cos (t)}{\frac{d}{d u} [e^{u}] \frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]}$

Step 1.3.4.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{t \to 0} \frac{\cos (t)}{e^{u} \frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]}$

Step 1.3.4.1.3

Replace all occurrences of $u$ with $3 t$ .

$lim_{t \to 0} \frac{\cos (t)}{e^{3 t} \frac{d}{d t} [3 t] + \frac{d}{d t} [- 1]}$

Step 1.3.4.2

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

$lim_{t \to 0} \frac{\cos (t)}{e^{3 t} (3 \frac{d}{d t} [t]) + \frac{d}{d t} [- 1]}$

Step 1.3.4.3

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

$lim_{t \to 0} \frac{\cos (t)}{e^{3 t} (3 \cdot 1) + \frac{d}{d t} [- 1]}$

Step 1.3.4.4

Multiply $3$ by $1$ .

$lim_{t \to 0} \frac{\cos (t)}{e^{3 t} \cdot 3 + \frac{d}{d t} [- 1]}$

Step 1.3.4.5

Move $3$ to the left of $e^{3 t}$ .

$lim_{t \to 0} \frac{\cos (t)}{3 e^{3 t} + \frac{d}{d t} [- 1]}$

Step 1.3.5

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

$lim_{t \to 0} \frac{\cos (t)}{3 e^{3 t} + 0}$

Step 1.3.6

Add $3 e^{3 t}$ and $0$ .

$lim_{t \to 0} \frac{\cos (t)}{3 e^{3 t}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{3} lim_{t \to 0} \frac{\cos (t)}{e^{3 t}}$

Step 2.2

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

$\frac{1}{3} \cdot \frac{lim_{t \to 0} \cos (t)}{lim_{t \to 0} e^{3 t}}$

Step 2.3

Move the limit inside the trig function because cosine is continuous.

$\frac{1}{3} \cdot \frac{\cos (lim_{t \to 0} t)}{lim_{t \to 0} e^{3 t}}$

Step 2.4

Move the limit into the exponent.

$\frac{1}{3} \cdot \frac{\cos (lim_{t \to 0} t)}{e^{lim_{t \to 0} 3 t}}$

Step 2.5

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

$\frac{1}{3} \cdot \frac{\cos (lim_{t \to 0} t)}{e^{3 lim_{t \to 0} t}}$

Step 3

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

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Step 3.1

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

$\frac{1}{3} \cdot \frac{\cos (0)}{e^{3 lim_{t \to 0} t}}$

Step 3.2

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

$\frac{1}{3} \cdot \frac{\cos (0)}{e^{3 \cdot 0}}$

Step 4

Simplify the answer.

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Step 4.1

Combine.

$\frac{1 \cos (0)}{3 e^{3 \cdot 0}}$

Step 4.2

Multiply $\cos (0)$ by $1$ .

$\frac{\cos (0)}{3 e^{3 \cdot 0}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Multiply $3$ by $0$ .

$\frac{\cos (0)}{3 e^{0}}$

Step 4.3.2

Anything raised to $0$ is $1$ .

$\frac{\cos (0)}{3 \cdot 1}$

Step 4.4

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{3 \cdot 1}$

Step 4.5

Multiply $3$ by $1$ .

$\frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$