Calculus Examples

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

Step 1

Multiply $\frac{\log_{4} (x)}{\log_{2} (x)}$ by $1$ .

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Move the limit inside the logarithm.

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Logarithm base $4$ of $1$ is $0$ .

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Move the limit inside the logarithm.

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

Step 2.1.3.2

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

$\frac{0}{\log_{2} (1)}$

Step 2.1.3.3

Logarithm base $2$ of $1$ is $0$ .

$\frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{x \to 1} \frac{\log_{4} (x)}{\log_{2} (x)} = lim_{x \to 1} \frac{\frac{d}{d x} [\log_{4} (x)]}{\frac{d}{d x} [\log_{2} (x)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

The derivative of $\log_{4} (x)$ with respect to $x$ is $\frac{1}{x \ln (4)}$ .

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

Step 2.3.3

The derivative of $\log_{2} (x)$ with respect to $x$ is $\frac{1}{x \ln (2)}$ .

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

Combine factors.

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Step 2.5.1

Combine $x$ and $\frac{1}{x \ln (4)}$ .

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

Step 2.5.2

Combine $\frac{x}{x \ln (4)}$ and $\ln (2)$ .

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

Step 2.6

Cancel the common factor of $x$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

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

Step 3

Evaluate the limit.

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Step 3.1

Evaluate the limit of $\frac{\ln (2)}{\ln (4)}$ which is constant as $x$ approaches $1$ .

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

Step 3.2

Simplify the answer.

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Step 3.2.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$\frac{\ln (2)}{\ln (2^{2})}$

Step 3.2.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

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

Step 3.2.3

Cancel the common factor of $\ln (2)$ .

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Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

Rewrite the expression.

$\frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$