Calculus Examples

$lim_{x \to - \infty} \frac{7 x + 2}{\sqrt{4 x^{2} + 5 x - 3}}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{7 x}{x} + \frac{2}{x}}{- \sqrt{\frac{4 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{\frac{4 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 3}{x^{2}}}}$

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.

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{\frac{4 x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 2.2.1.2

Divide $4$ by $1$ .

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{4 + \frac{5 x}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 2.2.2

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

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Step 2.2.2.1

Factor $x$ out of $5 x$ .

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{4 + \frac{x \cdot 5}{x^{2}} + \frac{- 3}{x^{2}}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $x$ out of $x^{2}$ .

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{4 + \frac{x \cdot 5}{x \cdot x} + \frac{- 3}{x^{2}}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{4 + \frac{x \cdot 5}{x \cdot x} + \frac{- 3}{x^{2}}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{4 + \frac{5}{x} + \frac{- 3}{x^{2}}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{7 + \frac{2}{x}}{- \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 2.3

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

$\frac{lim_{x \to - \infty} 7 + \frac{2}{x}}{lim_{x \to - \infty} - \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 2.4

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

$\frac{lim_{x \to - \infty} 7 + lim_{x \to - \infty} \frac{2}{x}}{lim_{x \to - \infty} - \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 2.5

Evaluate the limit of $7$ which is constant as $x$ approaches $- \infty$ .

$\frac{7 + lim_{x \to - \infty} \frac{2}{x}}{lim_{x \to - \infty} - \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 2.6

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

$\frac{7 + 2 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} - \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 3

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

$\frac{7 + 2 \cdot 0}{lim_{x \to - \infty} - \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{7 + 2 \cdot 0}{- lim_{x \to - \infty} \sqrt{4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 4.2

Move the limit under the radical sign.

$\frac{7 + 2 \cdot 0}{- \sqrt{lim_{x \to - \infty} 4 + \frac{5}{x} - \frac{3}{x^{2}}}}$

Step 4.3

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

$\frac{7 + 2 \cdot 0}{- \sqrt{lim_{x \to - \infty} 4 + lim_{x \to - \infty} \frac{5}{x} - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 4.4

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

$\frac{7 + 2 \cdot 0}{- \sqrt{4 + lim_{x \to - \infty} \frac{5}{x} - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 4.5

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

$\frac{7 + 2 \cdot 0}{- \sqrt{4 + 5 lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 5

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

$\frac{7 + 2 \cdot 0}{- \sqrt{4 + 5 \cdot 0 - lim_{x \to - \infty} \frac{3}{x^{2}}}}$

Step 6

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

$\frac{7 + 2 \cdot 0}{- \sqrt{4 + 5 \cdot 0 - 3 lim_{x \to - \infty} \frac{1}{x^{2}}}}$

Step 7

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

$\frac{7 + 2 \cdot 0}{- \sqrt{4 + 5 \cdot 0 - 3 \cdot 0}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $2$ by $0$ .

$\frac{7 + 0}{- \sqrt{4 + 5 \cdot 0 - 3 \cdot 0}}$

Step 8.1.2

Add $7$ and $0$ .

$\frac{7}{- \sqrt{4 + 5 \cdot 0 - 3 \cdot 0}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $5$ by $0$ .

$\frac{7}{- \sqrt{4 + 0 - 3 \cdot 0}}$

Step 8.2.2

Multiply $- 3$ by $0$ .

$\frac{7}{- \sqrt{4 + 0 + 0}}$

Step 8.2.3

Add $4 + 0$ and $0$ .

$\frac{7}{- \sqrt{4 + 0}}$

Step 8.2.4

Add $4$ and $0$ .

$\frac{7}{- \sqrt{4}}$

Step 8.2.5

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

$\frac{7}{- \sqrt{2^{2}}}$

Step 8.2.6

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

$\frac{7}{- 1 \cdot 2}$

Step 8.3

Multiply $- 1$ by $2$ .

$\frac{7}{- 2}$

Step 8.4

Move the negative in front of the fraction.

$- \frac{7}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{7}{2}$

Decimal Form:

$- 3.5$