Calculus Examples

$lim_{x \to 8} \frac{10}{5 - 15^{- \frac{x}{2}}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$10 lim_{x \to 8} \frac{1}{5 - 15^{- \frac{x}{2}}}$

Step 1.2

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

$10 \frac{lim_{x \to 8} 1}{lim_{x \to 8} 5 - 15^{- \frac{x}{2}}}$

Step 1.3

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

$10 \frac{1}{lim_{x \to 8} 5 - 15^{- \frac{x}{2}}}$

Step 1.4

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

$10 \frac{1}{lim_{x \to 8} 5 - lim_{x \to 8} 15^{- \frac{x}{2}}}$

Step 1.5

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$10 \frac{1}{5 - lim_{x \to 8} 15^{- \frac{x}{2}}}$

Step 1.6

Move the limit into the exponent.

$10 \frac{1}{5 - 15^{lim_{x \to 8} - \frac{x}{2}}}$

Step 1.7

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

$10 \frac{1}{5 - 15^{- lim_{x \to 8} \frac{x}{2}}}$

Step 1.8

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$10 \frac{1}{5 - 15^{- \frac{1}{2} lim_{x \to 8} x}}$

Step 2

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

$10 \frac{1}{5 - 15^{- \frac{1}{2} \cdot 8}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$10 \frac{1}{5 - 15^{\frac{- 1}{2} \cdot 8}}$

Step 3.1.1.2

Factor $2$ out of $8$ .

$10 \frac{1}{5 - 15^{\frac{- 1}{2} \cdot (2 (4))}}$

Step 3.1.1.3

Cancel the common factor.

$10 \frac{1}{5 - 15^{\frac{- 1}{2} \cdot (2 \cdot 4)}}$

Step 3.1.1.4

Rewrite the expression.

$10 \frac{1}{5 - 15^{- 1 \cdot 4}}$

Step 3.1.2

Multiply $- 1$ by $4$ .

$10 \frac{1}{5 - 15^{- 4}}$

Step 3.1.3

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

$10 \frac{1}{5 - \frac{1}{15^{4}}}$

Step 3.1.4

Raise $15$ to the power of $4$ .

$10 \frac{1}{5 - \frac{1}{50625}}$

Step 3.1.5

To write $5$ as a fraction with a common denominator, multiply by $\frac{50625}{50625}$ .

$10 \frac{1}{5 \cdot \frac{50625}{50625} - \frac{1}{50625}}$

Step 3.1.6

Combine $5$ and $\frac{50625}{50625}$ .

$10 \frac{1}{\frac{5 \cdot 50625}{50625} - \frac{1}{50625}}$

Step 3.1.7

Combine the numerators over the common denominator.

$10 \frac{1}{\frac{5 \cdot 50625 - 1}{50625}}$

Step 3.1.8

Simplify the numerator.

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Step 3.1.8.1

Multiply $5$ by $50625$ .

$10 \frac{1}{\frac{253125 - 1}{50625}}$

Step 3.1.8.2

Subtract $1$ from $253125$ .

$10 \frac{1}{\frac{253124}{50625}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$10 (1 (\frac{50625}{253124}))$

Step 3.3

Multiply $\frac{50625}{253124}$ by $1$ .

$10 (\frac{50625}{253124})$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Factor $2$ out of $10$ .

$2 (5) \frac{50625}{253124}$

Step 3.4.2

Factor $2$ out of $253124$ .

$2 \cdot 5 \frac{50625}{2 \cdot 126562}$

Step 3.4.3

Cancel the common factor.

$2 \cdot 5 \frac{50625}{2 \cdot 126562}$

Step 3.4.4

Rewrite the expression.

$5 (\frac{50625}{126562})$

Step 3.5

Combine $5$ and $\frac{50625}{126562}$ .

$\frac{5 \cdot 50625}{126562}$

Step 3.6

Multiply $5$ by $50625$ .

$\frac{253125}{126562}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{253125}{126562}$

Decimal Form:

$2.00000790 \dots$