Calculus Examples

lim x \to \infty \frac{x^{2}}{\sqrt{x^{2} + 1}}

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

Step 1

Divide the numerator and denominator by the highest power of

x

$x$ in the denominator, which is

x = \sqrt{x^{2}}

$x = \sqrt{x^{2}}$ .

lim x \to \infty \frac{\frac{x^{2}}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}}}

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of

x^{2}

$x^{2}$ and

x

$x$ .

lim x \to \infty \frac{x}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}}}

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

Step 2.2

Cancel the common factor of

x^{2}

$x^{2}$ .

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Step 2.2.1

Cancel the common factor.

lim x \to \infty \frac{x}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}}}

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

Step 2.2.2

Rewrite the expression.

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

Step 3

$x$ approaches

$\infty$ , the fraction

$\frac{1}{x^{2}}$ approaches

$0$ .

$lim_{x \to \infty} \frac{x}{\sqrt{1 + 0}}$

Step 4

Since its numerator is unbounded while its denominator approaches a constant number, the fraction

$\frac{x}{\sqrt{1 + 0}}$ approaches infinity.

$\infty$