Calculus Examples

$lim_{x \to - \infty} \frac{x}{\sqrt{x^{2} + 4}}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x = - \sqrt{x^{2}}$ .

$lim_{x \to - \infty} \frac{\frac{x}{x}}{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{4}{x^{2}}}}$

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{4}{x^{2}}}}$

Step 2.2

Cancel the common factor of $x^{2}$ .

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Step 2.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x^{2}}{x^{2}} + \frac{4}{x^{2}}}}$

Step 2.2.2

Rewrite the expression.

$lim_{x \to - \infty} \frac{1}{- \sqrt{1 + \frac{4}{x^{2}}}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} - \sqrt{1 + \frac{4}{x^{2}}}}$

Step 2.4

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

$\frac{1}{lim_{x \to - \infty} - \sqrt{1 + \frac{4}{x^{2}}}}$

Step 2.5

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{- lim_{x \to - \infty} \sqrt{1 + \frac{4}{x^{2}}}}$

Step 2.6

Move the limit under the radical sign.

$\frac{1}{- \sqrt{lim_{x \to - \infty} 1 + \frac{4}{x^{2}}}}$

Step 2.7

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

$\frac{1}{- \sqrt{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{4}{x^{2}}}}$

Step 2.8

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

$\frac{1}{- \sqrt{1 + lim_{x \to - \infty} \frac{4}{x^{2}}}}$

Step 2.9

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

$\frac{1}{- \sqrt{1 + 4 lim_{x \to - \infty} \frac{1}{x^{2}}}}$

Step 3

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

$\frac{1}{- \sqrt{1 + 4 \cdot 0}}$

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \sqrt{1 + 4 \cdot 0}}$

Step 4.1.2

Move the negative in front of the fraction.

$- \frac{1}{\sqrt{1 + 4 \cdot 0}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $4$ by $0$ .

$- \frac{1}{\sqrt{1 + 0}}$

Step 4.2.2

Add $1$ and $0$ .

$- \frac{1}{\sqrt{1}}$

Step 4.2.3

Any root of $1$ is $1$ .

$- \frac{1}{1}$

Step 4.3

Cancel the common factor of $1$ .

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Step 4.3.1

Cancel the common factor.

$- \frac{1}{1}$

Step 4.3.2

Rewrite the expression.

$- 1 \cdot 1$

Step 4.4

Multiply $- 1$ by $1$ .

$- 1$