Calculus Examples

$lim_{x \to \infty} \ln (4 x) - \ln (x + 1)$

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit inside the logarithm.

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

Step 2.2

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

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

Step 3

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\ln (4 \frac{1}{1 + 0})$

Step 6

Simplify the answer.

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Step 6.1

Add $1$ and $0$ .

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

Step 6.2

Cancel the common factor of $1$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

$\ln (4 \cdot 1)$

Step 6.3

Multiply $4$ by $1$ .

$\ln (4)$