Calculus Examples

$lim_{x \to 1} \frac{4^{x} - 4}{x^{2} - 1}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 1} 4^{x} - 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 4^{x} - lim_{x \to 1} 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2.1.2

Move the limit into the exponent.

$\frac{4^{lim_{x \to 1} x} - lim_{x \to 1} 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2.1.3

Evaluate the limit of $4$ which is constant as $x$ approaches $1$ .

$\frac{4^{lim_{x \to 1} x} - 1 \cdot 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{4^{1} - 1 \cdot 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Evaluate the exponent.

$\frac{4 - 1 \cdot 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2.3.1.2

Multiply $- 1$ by $4$ .

$\frac{4 - 4}{lim_{x \to 1} x^{2} - 1}$

Step 1.2.3.2

Subtract $4$ from $4$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1^{2} - 1 \cdot 1}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

One to any power is one.

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

Step 1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 1} \frac{4^{x} - 4}{x^{2} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [4^{x} - 4]}{\frac{d}{d x} [x^{2} - 1]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [4^{x} - 4]}{\frac{d}{d x} [x^{2} - 1]}$

Step 3.2

By the Sum Rule, the derivative of $4^{x} - 4$ with respect to $x$ is $\frac{d}{d x} [4^{x}] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [4^{x}] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [x^{2} - 1]}$

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $4$ .

$lim_{x \to 1} \frac{4^{x} \ln (4) + \frac{d}{d x} [- 4]}{\frac{d}{d x} [x^{2} - 1]}$

Step 3.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

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

Step 3.5

Add $4^{x} \ln (4)$ and $0$ .

$lim_{x \to 1} \frac{4^{x} \ln (4)}{\frac{d}{d x} [x^{2} - 1]}$

Step 3.6

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

$lim_{x \to 1} \frac{4^{x} \ln (4)}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.7

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

$lim_{x \to 1} \frac{4^{x} \ln (4)}{2 x + \frac{d}{d x} [- 1]}$

Step 3.8

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

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

Step 3.9

Add $2 x$ and $0$ .

$lim_{x \to 1} \frac{4^{x} \ln (4)}{2 x}$

Step 4

Move the term $\frac{\ln (4)}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (4)}{2} lim_{x \to 1} \frac{4^{x}}{x}$

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{\ln (4)}{2} \cdot \frac{lim_{x \to 1} 4^{x}}{lim_{x \to 1} x}$

Step 6

Move the limit into the exponent.

$\frac{\ln (4)}{2} \cdot \frac{4^{lim_{x \to 1} x}}{lim_{x \to 1} x}$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\ln (4)}{2} \cdot \frac{4^{1}}{lim_{x \to 1} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\ln (4)}{2} \cdot \frac{4^{1}}{1}$

Step 8

Simplify the answer.

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Step 8.1

Combine.

$\frac{\ln (4) \cdot 4^{1}}{2 \cdot 1}$

Step 8.2

Evaluate the exponent.

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

Step 8.3

Multiply $2$ by $1$ .

$\frac{\ln (4) \cdot 4}{2}$

Step 8.4

Cancel the common factor of $4$ and $2$ .

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Step 8.4.1

Factor $2$ out of $\ln (4) \cdot 4$ .

$\frac{2 (\ln (4) \cdot 2)}{2}$

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $2$ out of $2$ .

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

Step 8.4.2.2

Cancel the common factor.

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

Step 8.4.2.3

Rewrite the expression.

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

Step 8.4.2.4

Divide $\ln (4) \cdot 2$ by $1$ .

$\ln (4) \cdot 2$

Step 8.5

Move $2$ to the left of $\ln (4)$ .

$2 \ln (4)$