Calculus Examples

$lim_{x \to 0} \frac{3^{x} - 4^{x}}{x}$

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.

$\frac{lim_{x \to 0} 3^{x} - 4^{x}}{lim_{x \to 0} x}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 0} 3^{x} - lim_{x \to 0} 4^{x}}{lim_{x \to 0} x}$

Step 1.1.2.2

Move the limit into the exponent.

$\frac{3^{lim_{x \to 0} x} - lim_{x \to 0} 4^{x}}{lim_{x \to 0} x}$

Step 1.1.2.3

Move the limit into the exponent.

$\frac{3^{lim_{x \to 0} x} - 4^{lim_{x \to 0} x}}{lim_{x \to 0} x}$

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

$\frac{3^{0} - 4^{lim_{x \to 0} x}}{lim_{x \to 0} x}$

Step 1.1.2.4.2

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

$\frac{3^{0} - 4^{0}}{lim_{x \to 0} x}$

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - 4^{0}}{lim_{x \to 0} x}$

Step 1.1.2.5.1.2

Anything raised to $0$ is $1$ .

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

Step 1.1.2.5.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2.5.2

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.

$lim_{x \to 0} \frac{3^{x} - 4^{x}}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [3^{x} - 4^{x}]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [3^{x} - 4^{x}]}{\frac{d}{d x} [x]}$

Step 1.3.2

By the Sum Rule, the derivative of $3^{x} - 4^{x}$ with respect to $x$ is $\frac{d}{d x} [3^{x}] + \frac{d}{d x} [- 4^{x}]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [3^{x}] + \frac{d}{d x} [- 4^{x}]}{\frac{d}{d x} [x]}$

Step 1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$lim_{x \to 0} \frac{3^{x} \ln (3) + \frac{d}{d x} [- 4^{x}]}{\frac{d}{d x} [x]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 4^{x}]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 4^{x}$ with respect to $x$ is $- \frac{d}{d x} [4^{x}]$ .

$lim_{x \to 0} \frac{3^{x} \ln (3) - \frac{d}{d x} [4^{x}]}{\frac{d}{d x} [x]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{3^{x} \ln (3) - 4^{x} \ln (4)}{\frac{d}{d x} [x]}$

Step 1.3.5

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

$lim_{x \to 0} \frac{3^{x} \ln (3) - 4^{x} \ln (4)}{1}$

Step 1.4

Divide $3^{x} \ln (3) - 4^{x} \ln (4)$ by $1$ .

$lim_{x \to 0} 3^{x} \ln (3) - 4^{x} \ln (4)$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{x \to 0} 3^{x} \ln (3) - lim_{x \to 0} 4^{x} \ln (4)$

Step 2.2

Move the term $\ln (3)$ outside of the limit because it is constant with respect to $x$ .

$\ln (3) lim_{x \to 0} 3^{x} - lim_{x \to 0} 4^{x} \ln (4)$

Step 2.3

Move the limit into the exponent.

$\ln (3) \cdot 3^{lim_{x \to 0} x} - lim_{x \to 0} 4^{x} \ln (4)$

Step 2.4

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

$\ln (3) \cdot 3^{lim_{x \to 0} x} - \ln (4) lim_{x \to 0} 4^{x}$

Step 2.5

Move the limit into the exponent.

$\ln (3) \cdot 3^{lim_{x \to 0} x} - \ln (4) \cdot 4^{lim_{x \to 0} x}$

Step 3

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

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Step 3.1

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

$\ln (3) \cdot 3^{0} - \ln (4) \cdot 4^{lim_{x \to 0} x}$

Step 3.2

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

$\ln (3) \cdot 3^{0} - \ln (4) \cdot 4^{0}$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Anything raised to $0$ is $1$ .

$\ln (3) \cdot 1 - \ln (4) \cdot 4^{0}$

Step 4.1.2

Multiply $\ln (3)$ by $1$ .

$\ln (3) - \ln (4) \cdot 4^{0}$

Step 4.1.3

Anything raised to $0$ is $1$ .

$\ln (3) - \ln (4) \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$\ln (3) - \ln (4)$

Step 4.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{3}{4})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{3}{4})$

Decimal Form:

$- 0.28768207 \dots$