Calculus Examples

$lim_{x \to - \infty} \frac{\sqrt{7 + 4 x^{2}}}{2 x + 5}$

Step 1

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

$lim_{x \to - \infty} \frac{- \sqrt{\frac{7}{x^{2}} + \frac{4 x^{2}}{x^{2}}}}{\frac{2 x}{x} + \frac{5}{x}}$

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{7}{x^{2}} + \frac{4 x^{2}}{x^{2}}}}{2 + \frac{5}{x}}$

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{- \sqrt{\frac{7}{x^{2}} + \frac{4 x^{2}}{x^{2}}}}{2 + \frac{5}{x}}$

Step 2.2.2

Divide $4$ by $1$ .

$lim_{x \to - \infty} \frac{- \sqrt{\frac{7}{x^{2}} + 4}}{2 + \frac{5}{x}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} - \sqrt{\frac{7}{x^{2}} + 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 2.4

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

$\frac{- lim_{x \to - \infty} \sqrt{\frac{7}{x^{2}} + 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 2.5

Move the limit under the radical sign.

$\frac{- \sqrt{lim_{x \to - \infty} \frac{7}{x^{2}} + 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 2.6

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

$\frac{- \sqrt{lim_{x \to - \infty} \frac{7}{x^{2}} + lim_{x \to - \infty} 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 2.7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{7 lim_{x \to - \infty} \frac{1}{x^{2}} + lim_{x \to - \infty} 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 3

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

$\frac{- \sqrt{7 \cdot 0 + lim_{x \to - \infty} 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 4

Evaluate the limit.

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Step 4.1

Evaluate the limit of $4$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{7 \cdot 0 + 4}}{lim_{x \to - \infty} 2 + \frac{5}{x}}$

Step 4.2

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

$\frac{- \sqrt{7 \cdot 0 + 4}}{lim_{x \to - \infty} 2 + lim_{x \to - \infty} \frac{5}{x}}$

Step 4.3

Evaluate the limit of $2$ which is constant as $x$ approaches $- \infty$ .

$\frac{- \sqrt{7 \cdot 0 + 4}}{2 + lim_{x \to - \infty} \frac{5}{x}}$

Step 4.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{- \sqrt{7 \cdot 0 + 4}}{2 + 5 lim_{x \to - \infty} \frac{1}{x}}$

Step 5

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

$\frac{- \sqrt{7 \cdot 0 + 4}}{2 + 5 \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $7$ by $0$ .

$\frac{- \sqrt{0 + 4}}{2 + 5 \cdot 0}$

Step 6.1.2

Add $0$ and $4$ .

$\frac{- \sqrt{4}}{2 + 5 \cdot 0}$

Step 6.1.3

Rewrite $4$ as $2^{2}$ .

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

Step 6.1.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $5$ by $0$ .

$\frac{- 1 \cdot 2}{2 + 0}$

Step 6.2.2

Add $2$ and $0$ .

$\frac{- 1 \cdot 2}{2}$

Step 6.3

Multiply $- 1$ by $2$ .

$\frac{- 2}{2}$

Step 6.4

Divide $- 2$ by $2$ .

$- 1$