Calculus Examples

$lim_{x \to 8} \frac{7 + 5^{x} + 3^{- x}}{13 + 5^{x} + 9^{- x}}$

Step 1

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

$\frac{lim_{x \to 8} 7 + 5^{x} + 3^{- x}}{lim_{x \to 8} 13 + 5^{x} + 9^{- x}}$

Step 2

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

$\frac{lim_{x \to 8} 7 + lim_{x \to 8} 5^{x} + lim_{x \to 8} 3^{- x}}{lim_{x \to 8} 13 + 5^{x} + 9^{- x}}$

Step 3

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

$\frac{7 + lim_{x \to 8} 5^{x} + lim_{x \to 8} 3^{- x}}{lim_{x \to 8} 13 + 5^{x} + 9^{- x}}$

Step 4

Move the limit into the exponent.

$\frac{7 + 5^{lim_{x \to 8} x} + lim_{x \to 8} 3^{- x}}{lim_{x \to 8} 13 + 5^{x} + 9^{- x}}$

Step 5

Move the limit into the exponent.

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{lim_{x \to 8} - x}}{lim_{x \to 8} 13 + 5^{x} + 9^{- x}}$

Step 6

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

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{- lim_{x \to 8} x}}{lim_{x \to 8} 13 + 5^{x} + 9^{- x}}$

Step 7

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

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{- lim_{x \to 8} x}}{lim_{x \to 8} 13 + lim_{x \to 8} 5^{x} + lim_{x \to 8} 9^{- x}}$

Step 8

Evaluate the limit of $13$ which is constant as $x$ approaches $8$ .

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{- lim_{x \to 8} x}}{13 + lim_{x \to 8} 5^{x} + lim_{x \to 8} 9^{- x}}$

Step 9

Move the limit into the exponent.

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{- lim_{x \to 8} x}}{13 + 5^{lim_{x \to 8} x} + lim_{x \to 8} 9^{- x}}$

Step 10

Move the limit into the exponent.

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{- lim_{x \to 8} x}}{13 + 5^{lim_{x \to 8} x} + 9^{lim_{x \to 8} - x}}$

Step 11

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

$\frac{7 + 5^{lim_{x \to 8} x} + 3^{- lim_{x \to 8} x}}{13 + 5^{lim_{x \to 8} x} + 9^{- lim_{x \to 8} x}}$

Step 12

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

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Step 12.1

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

$\frac{7 + 5^{8} + 3^{- lim_{x \to 8} x}}{13 + 5^{lim_{x \to 8} x} + 9^{- lim_{x \to 8} x}}$

Step 12.2

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

$\frac{7 + 5^{8} + 3^{- 8}}{13 + 5^{lim_{x \to 8} x} + 9^{- lim_{x \to 8} x}}$

Step 12.3

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

$\frac{7 + 5^{8} + 3^{- 8}}{13 + 5^{8} + 9^{- lim_{x \to 8} x}}$

Step 12.4

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

$\frac{7 + 5^{8} + 3^{- 8}}{13 + 5^{8} + 9^{- 8}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Raise $5$ to the power of $8$ .

$\frac{7 + 390625 + 3^{- 8}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.2

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

$\frac{7 + 390625 + \frac{1}{3^{8}}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.3

Raise $3$ to the power of $8$ .

$\frac{7 + 390625 + \frac{1}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.4

Add $7$ and $390625$ .

$\frac{390632 + \frac{1}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.5

To write $390632$ as a fraction with a common denominator, multiply by $\frac{6561}{6561}$ .

$\frac{390632 \cdot \frac{6561}{6561} + \frac{1}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.6

Combine $390632$ and $\frac{6561}{6561}$ .

$\frac{\frac{390632 \cdot 6561}{6561} + \frac{1}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.7

Combine the numerators over the common denominator.

$\frac{\frac{390632 \cdot 6561 + 1}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.8

Simplify the numerator.

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Step 13.1.8.1

Multiply $390632$ by $6561$ .

$\frac{\frac{2562936552 + 1}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.1.8.2

Add $2562936552$ and $1$ .

$\frac{\frac{2562936553}{6561}}{13 + 5^{8} + 9^{- 8}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Raise $5$ to the power of $8$ .

$\frac{\frac{2562936553}{6561}}{13 + 390625 + 9^{- 8}}$

Step 13.2.2

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

$\frac{\frac{2562936553}{6561}}{13 + 390625 + \frac{1}{9^{8}}}$

Step 13.2.3

Raise $9$ to the power of $8$ .

$\frac{\frac{2562936553}{6561}}{13 + 390625 + \frac{1}{43046721}}$

Step 13.2.4

Add $13$ and $390625$ .

$\frac{\frac{2562936553}{6561}}{390638 + \frac{1}{43046721}}$

Step 13.2.5

To write $390638$ as a fraction with a common denominator, multiply by $\frac{43046721}{43046721}$ .

$\frac{\frac{2562936553}{6561}}{390638 \cdot \frac{43046721}{43046721} + \frac{1}{43046721}}$

Step 13.2.6

Combine $390638$ and $\frac{43046721}{43046721}$ .

$\frac{\frac{2562936553}{6561}}{\frac{390638 \cdot 43046721}{43046721} + \frac{1}{43046721}}$

Step 13.2.7

Combine the numerators over the common denominator.

$\frac{\frac{2562936553}{6561}}{\frac{390638 \cdot 43046721 + 1}{43046721}}$

Step 13.2.8

Simplify the numerator.

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Step 13.2.8.1

Multiply $390638$ by $43046721$ .

$\frac{\frac{2562936553}{6561}}{\frac{16815684997998 + 1}{43046721}}$

Step 13.2.8.2

Add $16815684997998$ and $1$ .

$\frac{\frac{2562936553}{6561}}{\frac{16815684997999}{43046721}}$

Step 13.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{2562936553}{6561} \cdot \frac{43046721}{16815684997999}$

Step 13.4

Cancel the common factor of $6561$ .

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Step 13.4.1

Factor $6561$ out of $43046721$ .

$\frac{2562936553}{6561} \cdot \frac{6561 (6561)}{16815684997999}$

Step 13.4.2

Cancel the common factor.

$\frac{2562936553}{6561} \cdot \frac{6561 \cdot 6561}{16815684997999}$

Step 13.4.3

Rewrite the expression.

$2562936553 (\frac{6561}{16815684997999})$

Step 13.5

Combine $2562936553$ and $\frac{6561}{16815684997999}$ .

$\frac{2562936553 \cdot 6561}{16815684997999}$

Step 13.6

Multiply $2562936553$ by $6561$ .

$\frac{16815426724233}{16815684997999}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{16815426724233}{16815684997999}$

Decimal Form:

$0.99998464 \dots$