Calculus Examples

$lim_{x \to - \infty} \frac{\sqrt{5 x^{2} - 2}}{x + 3}$

Step 1

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

$lim_{x \to - \infty} \frac{- \sqrt{\frac{5 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}}{\frac{x}{x} + \frac{3}{x}}$

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{5 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}}{1 + \frac{3}{x}}$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{5 x^{2}}{x^{2}} + \frac{- 2}{x^{2}}}}{1 + \frac{3}{x}}$

Step 2.2.1.2

Divide $5$ by $1$ .

$lim_{x \to - \infty} \frac{- \sqrt{5 + \frac{- 2}{x^{2}}}}{1 + \frac{3}{x}}$

Step 2.2.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{- \sqrt{5 - \frac{2}{x^{2}}}}{1 + \frac{3}{x}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} - \sqrt{5 - \frac{2}{x^{2}}}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 2.4

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

$\frac{- lim_{x \to - \infty} \sqrt{5 - \frac{2}{x^{2}}}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 2.5

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} 5 - \frac{2}{x^{2}}}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 2.6

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

$\frac{- \sqrt{lim_{x \to - \infty} 5 - lim_{x \to - \infty} \frac{2}{x^{2}}}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 2.7

Evaluate the limit of $5$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{5 - lim_{x \to - \infty} \frac{2}{x^{2}}}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 2.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{5 - 2 lim_{x \to - \infty} \frac{1}{x^{2}}}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 3

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

$\frac{- \sqrt{5 - 2 \cdot 0}}{lim_{x \to - \infty} 1 + \frac{3}{x}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{- \sqrt{5 - 2 \cdot 0}}{lim_{x \to - \infty} 1 + lim_{x \to - \infty} \frac{3}{x}}$

Step 4.2

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

$\frac{- \sqrt{5 - 2 \cdot 0}}{1 + lim_{x \to - \infty} \frac{3}{x}}$

Step 4.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{5 - 2 \cdot 0}}{1 + 3 lim_{x \to - \infty} \frac{1}{x}}$

Step 5

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

$\frac{- \sqrt{5 - 2 \cdot 0}}{1 + 3 \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 2$ by $0$ .

$\frac{- \sqrt{5 + 0}}{1 + 3 \cdot 0}$

Step 6.1.2

Add $5$ and $0$ .

$\frac{- \sqrt{5}}{1 + 3 \cdot 0}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $3$ by $0$ .

$\frac{- \sqrt{5}}{1 + 0}$

Step 6.2.2

Add $1$ and $0$ .

$\frac{- \sqrt{5}}{1}$

Step 6.3

Divide $- \sqrt{5}$ by $1$ .

$- \sqrt{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{5}$

Decimal Form:

$- 2.23606797 \dots$