Calculus Examples

$lim_{x \to 8} {(1 + 2 x)}^{\frac{1}{2 \ln (x)}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + 2 x)}^{\frac{1}{2 \ln (x)}}$ as $e^{\ln ({(1 + 2 x)}^{\frac{1}{2 \ln (x)}})}$ .

$lim_{x \to 8} e^{\ln ({(1 + 2 x)}^{\frac{1}{2 \ln (x)}})}$

Step 1.2

Expand $\ln ({(1 + 2 x)}^{\frac{1}{2 \ln (x)}})$ by moving $\frac{1}{2 \ln (x)}$ outside the logarithm.

$lim_{x \to 8} e^{\frac{1}{2 \ln (x)} \ln (1 + 2 x)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 8} \frac{1}{2 \ln (x)} \ln (1 + 2 x)}$

Step 2.2

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

$e^{lim_{x \to 8} \frac{1}{2 \ln (x)} \cdot lim_{x \to 8} \ln (1 + 2 x)}$

Step 2.3

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

$e^{\frac{1}{2} lim_{x \to 8} \frac{1}{\ln (x)} \cdot lim_{x \to 8} \ln (1 + 2 x)}$

Step 2.4

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

$e^{\frac{1}{2} \cdot \frac{lim_{x \to 8} 1}{lim_{x \to 8} \ln (x)} \cdot lim_{x \to 8} \ln (1 + 2 x)}$

Step 2.5

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

$e^{\frac{1}{2} \cdot \frac{1}{lim_{x \to 8} \ln (x)} \cdot lim_{x \to 8} \ln (1 + 2 x)}$

Step 2.6

Move the limit inside the logarithm.

$e^{\frac{1}{2} \cdot \frac{1}{\ln (lim_{x \to 8} x)} \cdot lim_{x \to 8} \ln (1 + 2 x)}$

Step 2.7

Move the limit inside the logarithm.

$e^{\frac{1}{2} \cdot \frac{1}{\ln (lim_{x \to 8} x)} \cdot \ln (lim_{x \to 8} 1 + 2 x)}$

Step 2.8

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

$e^{\frac{1}{2} \cdot \frac{1}{\ln (lim_{x \to 8} x)} \cdot \ln (lim_{x \to 8} 1 + lim_{x \to 8} 2 x)}$

Step 2.9

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

$e^{\frac{1}{2} \cdot \frac{1}{\ln (lim_{x \to 8} x)} \cdot \ln (1 + lim_{x \to 8} 2 x)}$

Step 2.10

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

$e^{\frac{1}{2} \cdot \frac{1}{\ln (lim_{x \to 8} x)} \cdot \ln (1 + 2 lim_{x \to 8} x)}$

Step 3

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

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Step 3.1

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

$e^{\frac{1}{2} \cdot \frac{1}{\ln (8)} \cdot \ln (1 + 2 lim_{x \to 8} x)}$

Step 3.2

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

$e^{\frac{1}{2} \cdot \frac{1}{\ln (8)} \cdot \ln (1 + 2 \cdot 8)}$

Step 4

Simplify the answer.

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Step 4.1

Multiply $\frac{1}{2}$ by $\frac{1}{\ln (8)}$ .

$e^{\frac{1}{2 \ln (8)} \cdot \ln (1 + 2 \cdot 8)}$

Step 4.2

Multiply $2$ by $8$ .

$e^{\frac{1}{2 \ln (8)} \cdot \ln (1 + 16)}$

Step 4.3

Add $1$ and $16$ .

$e^{\frac{1}{2 \ln (8)} \cdot \ln (17)}$

Step 4.4

Combine $\frac{1}{2 \ln (8)}$ and $\ln (17)$ .

$e^{\frac{\ln (17)}{2 \ln (8)}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{\ln (17)}{2 \ln (8)}}$

Decimal Form:

$1.97633438 \dots$