Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

Combine $\frac{1}{x^{2}}$ and $\ln (x)$ .

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

As log approaches infinity, the value goes to $\infty$ .

$e^{\frac{\infty}{lim_{x \to \infty} x^{2}}}$

Step 3.1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$e^{\frac{\infty}{\infty}}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

Step 3.2

Since $\frac{\infty}{\infty}$ 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.

$lim_{x \to \infty} \frac{\ln (x)}{x^{2}} = lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x^{2}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.2

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

$e^{lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{d}{d x} [x^{2}]}}$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{lim_{x \to \infty} \frac{\frac{1}{x}}{2 x}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{lim_{x \to \infty} \frac{1}{x} \cdot \frac{1}{2 x}}$

Step 3.5

Combine factors.

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Step 3.5.1

Multiply $\frac{1}{x}$ by $\frac{1}{2 x}$ .

$e^{lim_{x \to \infty} \frac{1}{x \cdot 2 x}}$

Step 3.5.2

Raise $x$ to the power of $1$ .

$e^{lim_{x \to \infty} \frac{1}{2 x^{1} x}}$

Step 3.5.3

Raise $x$ to the power of $1$ .

$e^{lim_{x \to \infty} \frac{1}{2 x^{1} x^{1}}}$

Step 3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.5

Add $1$ and $1$ .

$e^{lim_{x \to \infty} \frac{1}{2 x^{2}}}$

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{1}{2} lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$e^{\frac{1}{2} \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Multiply $\frac{1}{2}$ by $0$ .

$e^{0}$

Step 6.2

Anything raised to $0$ is $1$ .

$1$