Calculus Examples

$lim_{x \to 0} \frac{2^{x} - 1}{x}$

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.

$\frac{lim_{x \to 0} 2^{x} - 1}{lim_{x \to 0} x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{lim_{x \to 0} 2^{x} - lim_{x \to 0} 1}{lim_{x \to 0} x}$

Step 1.1.2.1.2

Move the limit into the exponent.

$\frac{2^{lim_{x \to 0} x} - lim_{x \to 0} 1}{lim_{x \to 0} x}$

Step 1.1.2.1.3

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

$\frac{2^{lim_{x \to 0} x} - 1 \cdot 1}{lim_{x \to 0} x}$

Step 1.1.2.2

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

$\frac{2^{0} - 1 \cdot 1}{lim_{x \to 0} x}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} x}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} x}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} x}$

Step 1.1.3

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

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{2^{x} - 1}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [2^{x} - 1]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [2^{x} - 1]}{\frac{d}{d x} [x]}$

Step 1.3.2

By the Sum Rule, the derivative of $2^{x} - 1$ with respect to $x$ is $\frac{d}{d x} [2^{x}] + \frac{d}{d x} [- 1]$ .

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

Step 1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

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

Step 1.3.4

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

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

Step 1.3.5

Add $2^{x} \ln (2)$ and $0$ .

$lim_{x \to 0} \frac{2^{x} \ln (2)}{\frac{d}{d x} [x]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{2^{x} \ln (2)}{1}$

Step 1.4

Divide $2^{x} \ln (2)$ by $1$ .

$lim_{x \to 0} 2^{x} \ln (2)$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\ln (2)$ outside of the limit because it is constant with respect to $x$ .

$\ln (2) lim_{x \to 0} 2^{x}$

Step 2.2

Move the limit into the exponent.

$\ln (2) \cdot 2^{lim_{x \to 0} x}$

Step 3

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

$\ln (2) \cdot 2^{0}$

Step 4

Simplify the answer.

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Step 4.1

Anything raised to $0$ is $1$ .

$\ln (2) \cdot 1$

Step 4.2

Multiply $\ln (2)$ by $1$ .

$\ln (2)$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\ln (2)$

Decimal Form:

$0.69314718 \dots$