Calculus Examples

$lim_{x \to - \infty} \frac{4 x + 8}{5 x}$

Step 1

Move the term $\frac{1}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} lim_{x \to - \infty} \frac{4 x + 8}{x}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

$\frac{1}{5} lim_{x \to - \infty} \frac{\frac{4 x}{x} + \frac{8}{x}}{\frac{x}{x}}$

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

$\frac{1}{5} lim_{x \to - \infty} \frac{\frac{4 x}{x} + \frac{8}{x}}{\frac{x}{x}}$

Step 3.1.2

Divide $4$ by $1$ .

$\frac{1}{5} lim_{x \to - \infty} \frac{4 + \frac{8}{x}}{\frac{x}{x}}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$\frac{1}{5} lim_{x \to - \infty} \frac{4 + \frac{8}{x}}{\frac{x}{x}}$

Step 3.2.2

Rewrite the expression.

$\frac{1}{5} lim_{x \to - \infty} \frac{4 + \frac{8}{x}}{1}$

Step 3.3

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

$\frac{1}{5} \cdot \frac{lim_{x \to - \infty} 4 + \frac{8}{x}}{lim_{x \to - \infty} 1}$

Step 3.4

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

$\frac{1}{5} \cdot \frac{lim_{x \to - \infty} 4 + lim_{x \to - \infty} \frac{8}{x}}{lim_{x \to - \infty} 1}$

Step 3.5

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

$\frac{1}{5} \cdot \frac{4 + lim_{x \to - \infty} \frac{8}{x}}{lim_{x \to - \infty} 1}$

Step 3.6

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{5} \cdot \frac{4 + 8 lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}$

Step 4

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

$\frac{1}{5} \cdot \frac{4 + 8 \cdot 0}{lim_{x \to - \infty} 1}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{1}{5} \cdot \frac{4 + 8 \cdot 0}{1}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Divide $4 + 8 \cdot 0$ by $1$ .

$\frac{1}{5} (4 + 8 \cdot 0)$

Step 5.2.2

Multiply $8$ by $0$ .

$\frac{1}{5} (4 + 0)$

Step 5.2.3

Add $4$ and $0$ .

$\frac{1}{5} \cdot 4$

Step 5.2.4

Combine $\frac{1}{5}$ and $4$ .

$\frac{4}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$