Calculus Examples

$lim_{x \to 8} \frac{4 x}{20 x + 1}$

Step 1

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

$4 lim_{x \to 8} \frac{x}{20 x + 1}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$4 \frac{lim_{x \to 8} x}{lim_{x \to 8} 20 x + 1}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$4 \frac{lim_{x \to 8} x}{lim_{x \to 8} 20 x + lim_{x \to 8} 1}$

Step 4

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$4 \frac{lim_{x \to 8} x}{20 lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$4 \frac{lim_{x \to 8} x}{20 lim_{x \to 8} x + 1}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$4 \frac{8}{20 lim_{x \to 8} x + 1}$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$4 \frac{8}{20 \cdot 8 + 1}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Multiply $20$ by $8$ .

$4 \frac{8}{160 + 1}$

Step 7.1.2

Add $160$ and $1$ .

$4 (\frac{8}{161})$

Step 7.2

Multiply $4 (\frac{8}{161})$ .

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Step 7.2.1

Combine $4$ and $\frac{8}{161}$ .

$\frac{4 \cdot 8}{161}$

Step 7.2.2

Multiply $4$ by $8$ .

$\frac{32}{161}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{161}$

Decimal Form:

$0.19875776 \dots$