Calculus Examples

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

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.

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

Step 1.1.2

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

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

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 $0$ .

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

Step 1.1.3.1.2

Move the limit into the exponent.

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

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

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

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

Anything raised to $0$ is $1$ .

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

Step 1.1.3.3.1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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

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 $0$ .

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

Step 2.2

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

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

Step 2.3

Move the limit into the exponent.

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Anything raised to $0$ is $1$ .

$\frac{1}{1}$

Step 4.2

Divide $1$ by $1$ .

$1$