Calculus Examples

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

Step 1

Change the two-sided limit into a right sided limit.

$lim_{x \to \infty^{+}} {(1 - \frac{1}{x})}^{x}$

Step 2

Combine terms.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$lim_{x \to \infty^{+}} {(\frac{x}{x} - \frac{1}{x})}^{x}$

Step 2.2

Combine the numerators over the common denominator.

$lim_{x \to \infty^{+}} {(\frac{x - 1}{x})}^{x}$

Step 3

Use the properties of logarithms to simplify the limit.

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Step 3.1

Rewrite ${(\frac{x - 1}{x})}^{x}$ as $e^{\ln ({(\frac{x - 1}{x})}^{x})}$ .

$lim_{x \to \infty^{+}} e^{\ln ({(\frac{x - 1}{x})}^{x})}$

Step 3.2

Expand $\ln ({(\frac{x - 1}{x})}^{x})$ by moving $x$ outside the logarithm.

$lim_{x \to \infty^{+}} e^{x \ln (\frac{x - 1}{x})}$

Step 4

Because there are no values to the right of $\infty$ in the domain of $e^{x \ln (\frac{x - 1}{x})}$ , the limit does not exist.

$Does not exist$