Calculus Examples

$lim_{x \to 8} \frac{\frac{2}{2 x + 7}}{10 x}$

Step 1

Move the term $\frac{1}{10}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{10} lim_{x \to 8} \frac{\frac{2}{2 x + 7}}{x}$

Step 2

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

$\frac{1}{10} \cdot \frac{lim_{x \to 8} \frac{2}{2 x + 7}}{lim_{x \to 8} x}$

Step 3

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

$\frac{1}{10} \cdot \frac{2 lim_{x \to 8} \frac{1}{2 x + 7}}{lim_{x \to 8} x}$

Step 4

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

$\frac{1}{10} \cdot \frac{2 \frac{lim_{x \to 8} 1}{lim_{x \to 8} 2 x + 7}}{lim_{x \to 8} x}$

Step 5

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

$\frac{1}{10} \cdot \frac{2 \frac{1}{lim_{x \to 8} 2 x + 7}}{lim_{x \to 8} x}$

Step 6

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

$\frac{1}{10} \cdot \frac{2 \frac{1}{lim_{x \to 8} 2 x + lim_{x \to 8} 7}}{lim_{x \to 8} x}$

Step 7

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

$\frac{1}{10} \cdot \frac{2 \frac{1}{2 lim_{x \to 8} x + lim_{x \to 8} 7}}{lim_{x \to 8} x}$

Step 8

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{1}{10} \cdot \frac{2 \frac{1}{2 lim_{x \to 8} x + 7}}{lim_{x \to 8} x}$

Step 9

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

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Step 9.1

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

$\frac{1}{10} \cdot \frac{2 \frac{1}{2 \cdot 8 + 7}}{lim_{x \to 8} x}$

Step 9.2

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

$\frac{1}{10} \cdot \frac{2 \frac{1}{2 \cdot 8 + 7}}{8}$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $2$ .

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Step 10.1.1

Factor $2$ out of $10$ .

$\frac{1}{2 (5)} \cdot \frac{2 \frac{1}{2 \cdot 8 + 7}}{8}$

Step 10.1.2

Cancel the common factor.

$\frac{1}{2 \cdot 5} \cdot \frac{2 \frac{1}{2 \cdot 8 + 7}}{8}$

Step 10.1.3

Rewrite the expression.

$\frac{1}{5} \cdot \frac{\frac{1}{2 \cdot 8 + 7}}{8}$

Step 10.2

Multiply $2$ by $8$ .

$\frac{1}{5} \cdot \frac{\frac{1}{16 + 7}}{8}$

Step 10.3

Multiply $\frac{1}{5}$ by $\frac{\frac{1}{16 + 7}}{8}$ .

$\frac{\frac{1}{16 + 7}}{5 \cdot 8}$

Step 10.4

Multiply $5$ by $8$ .

$\frac{\frac{1}{16 + 7}}{40}$

Step 10.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{16 + 7} \cdot \frac{1}{40}$

Step 10.6

Add $16$ and $7$ .

$\frac{1}{23} \cdot \frac{1}{40}$

Step 10.7

Multiply $\frac{1}{23} \cdot \frac{1}{40}$ .

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Step 10.7.1

Multiply $\frac{1}{23}$ by $\frac{1}{40}$ .

$\frac{1}{23 \cdot 40}$

Step 10.7.2

Multiply $23$ by $40$ .

$\frac{1}{920}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{920}$

Decimal Form:

$0.00108695 \dots$