Calculus Examples

$lim_{x \to \infty} {(1 + \frac{1}{x})}^{x^{2}}$

Step 1

Combine terms.

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Step 1.1

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

$lim_{x \to \infty} {(\frac{x}{x} + \frac{1}{x})}^{x^{2}}$

Step 1.2

Combine the numerators over the common denominator.

$lim_{x \to \infty} {(\frac{x + 1}{x})}^{x^{2}}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(\frac{x + 1}{x})}^{x^{2}}$ as $e^{\ln ({(\frac{x + 1}{x})}^{x^{2}})}$ .

$lim_{x \to \infty} e^{\ln ({(\frac{x + 1}{x})}^{x^{2}})}$

Step 2.2

Expand $\ln ({(\frac{x + 1}{x})}^{x^{2}})$ by moving $x^{2}$ outside the logarithm.

$lim_{x \to \infty} e^{x^{2} \ln (\frac{x + 1}{x})}$

Step 3

Since the exponent $x^{2} \ln (\frac{x + 1}{x})$ approaches $\infty$ , the quantity $e^{x^{2} \ln (\frac{x + 1}{x})}$ approaches $\infty$ .

$\infty$