Calculus Examples

lim x \to 8 {(1 + \frac{a}{x})}^{b x}

$lim_{x \to 8} {(1 + \frac{a}{x})}^{b x}$

Step 1

Combine terms.

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Step 1.1

Write

1

$1$ as a fraction with a common denominator.

lim x \to 8 {(\frac{x}{x} + \frac{a}{x})}^{b x}

$lim_{x \to 8} {(\frac{x}{x} + \frac{a}{x})}^{b x}$

Step 1.2

Combine the numerators over the common denominator.

lim x \to 8 {(\frac{x + a}{x})}^{b x}

$lim_{x \to 8} {(\frac{x + a}{x})}^{b x}$

lim x \to 8 {(\frac{x + a}{x})}^{b x}

$lim_{x \to 8} {(\frac{x + a}{x})}^{b x}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite

{(\frac{x + a}{x})}^{b x}

${(\frac{x + a}{x})}^{b x}$ as

e^{ln ({(\frac{x + a}{x})}^{b x})}

$e^{\ln ({(\frac{x + a}{x})}^{b x})}$ .

$lim_{x \to 8} e^{\ln ({(\frac{x + a}{x})}^{b x})}$

Step 2.2

Expand

$\ln ({(\frac{x + a}{x})}^{b x})$ by moving

$b x$ outside the logarithm.

$lim_{x \to 8} e^{b x \ln (\frac{x + a}{x})}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$e^{lim_{x \to 8} b x \ln (\frac{x + a}{x})}$

Step 3.2

Move the term

$b$ outside of the limit because it is constant with respect to

$x$ .

$e^{b lim_{x \to 8} x \ln (\frac{x + a}{x})}$

Step 3.3

Split the limit using the Product of Limits Rule on the limit as

$x$ approaches

$8$ .

$e^{b lim_{x \to 8} x \cdot lim_{x \to 8} \ln (\frac{x + a}{x})}$

Step 3.4

Move the limit inside the logarithm.

$e^{b lim_{x \to 8} x \cdot \ln (lim_{x \to 8} \frac{x + a}{x})}$

Step 3.5

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$8$ .

$e^{b lim_{x \to 8} x \cdot \ln (\frac{lim_{x \to 8} x + a}{lim_{x \to 8} x})}$

Step 3.6

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$8$ .

$e^{b lim_{x \to 8} x \cdot \ln (\frac{lim_{x \to 8} x + lim_{x \to 8} a}{lim_{x \to 8} x})}$

Step 3.7

Evaluate the limit of

$a$ which is constant as

$x$ approaches

$8$ .

$e^{b lim_{x \to 8} x \cdot \ln (\frac{lim_{x \to 8} x + a}{lim_{x \to 8} x})}$

Step 4

Evaluate the limits by plugging in

$8$ for all occurrences of

$x$ .

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Step 4.1

Evaluate the limit of

$x$ by plugging in

$8$ for

$x$ .

$e^{b \cdot 8 \cdot \ln (\frac{lim_{x \to 8} x + a}{lim_{x \to 8} x})}$

Step 4.2

Evaluate the limit of

$x$ by plugging in

$8$ for

$x$ .

$e^{b \cdot 8 \cdot \ln (\frac{8 + a}{lim_{x \to 8} x})}$

Step 4.3

Evaluate the limit of

$x$ by plugging in

$8$ for

$x$ .

$e^{b \cdot 8 \cdot \ln (\frac{8 + a}{8})}$

Step 5

Move

$8$ to the left of

$b$ .

$e^{8 b \ln (\frac{8 + a}{8})}$