Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Evaluate the limit of $10$ which is constant as $x$ approaches $- 8$ .

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

Step 4

Move the limit into the exponent.

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

Step 5

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

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

Step 6

Evaluate the limit of $10$ which is constant as $x$ approaches $- 8$ .

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

Step 7

Move the limit into the exponent.

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

Step 8

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

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

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 $x$ .

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

Step 9.2

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

$\frac{10 - 2^{- 8}}{10 + 2^{8}}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10 - \frac{1}{2^{8}}}{10 + 2^{8}}$

Step 10.1.2

Raise $2$ to the power of $8$ .

$\frac{10 - \frac{1}{256}}{10 + 2^{8}}$

Step 10.1.3

To write $10$ as a fraction with a common denominator, multiply by $\frac{256}{256}$ .

$\frac{10 \cdot \frac{256}{256} - \frac{1}{256}}{10 + 2^{8}}$

Step 10.1.4

Combine $10$ and $\frac{256}{256}$ .

$\frac{\frac{10 \cdot 256}{256} - \frac{1}{256}}{10 + 2^{8}}$

Step 10.1.5

Combine the numerators over the common denominator.

$\frac{\frac{10 \cdot 256 - 1}{256}}{10 + 2^{8}}$

Step 10.1.6

Simplify the numerator.

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Step 10.1.6.1

Multiply $10$ by $256$ .

$\frac{\frac{2560 - 1}{256}}{10 + 2^{8}}$

Step 10.1.6.2

Subtract $1$ from $2560$ .

$\frac{\frac{2559}{256}}{10 + 2^{8}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $2$ to the power of $8$ .

$\frac{\frac{2559}{256}}{10 + 256}$

Step 10.2.2

Add $10$ and $256$ .

$\frac{\frac{2559}{256}}{266}$

Step 10.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{2559}{256} \cdot \frac{1}{266}$

Step 10.4

Multiply $\frac{2559}{256} \cdot \frac{1}{266}$ .

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Step 10.4.1

Multiply $\frac{2559}{256}$ by $\frac{1}{266}$ .

$\frac{2559}{256 \cdot 266}$

Step 10.4.2

Multiply $256$ by $266$ .

$\frac{2559}{68096}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2559}{68096}$

Decimal Form:

$0.03757929 \dots$