Calculus Examples

$lim_{x \to 8} {(\frac{16 x}{16 x + 6})}^{5 x}$

Step 1

Cancel the common factor of $16$ and $16 x + 6$ .

Tap for more steps...

Step 1.1

Factor $2$ out of $16 x$ .

$lim_{x \to 8} {(\frac{2 (8 x)}{16 x + 6})}^{5 x}$

Step 1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1

Factor $2$ out of $16 x$ .

$lim_{x \to 8} {(\frac{2 (8 x)}{2 (8 x) + 6})}^{5 x}$

Step 1.2.2

Factor $2$ out of $6$ .

$lim_{x \to 8} {(\frac{2 (8 x)}{2 (8 x) + 2 (3)})}^{5 x}$

Step 1.2.3

Factor $2$ out of $2 (8 x) + 2 (3)$ .

$lim_{x \to 8} {(\frac{2 (8 x)}{2 (8 x + 3)})}^{5 x}$

Step 1.2.4

Cancel the common factor.

$lim_{x \to 8} {(\frac{2 (8 x)}{2 (8 x + 3)})}^{5 x}$

Step 1.2.5

Rewrite the expression.

$lim_{x \to 8} {(\frac{8 x}{8 x + 3})}^{5 x}$

Step 2

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 2.1

Rewrite ${(\frac{8 x}{8 x + 3})}^{5 x}$ as $e^{\ln ({(\frac{8 x}{8 x + 3})}^{5 x})}$ .

$lim_{x \to 8} e^{\ln ({(\frac{8 x}{8 x + 3})}^{5 x})}$

Step 2.2

Expand $\ln ({(\frac{8 x}{8 x + 3})}^{5 x})$ by moving $5 x$ outside the logarithm.

$lim_{x \to 8} e^{5 x \ln (\frac{8 x}{8 x + 3})}$

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Move the limit into the exponent.

$e^{lim_{x \to 8} 5 x \ln (\frac{8 x}{8 x + 3})}$

Step 3.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$e^{5 lim_{x \to 8} x \ln (\frac{8 x}{8 x + 3})}$

Step 3.3

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

$e^{5 lim_{x \to 8} x \cdot lim_{x \to 8} \ln (\frac{8 x}{8 x + 3})}$

Step 3.4

Move the limit inside the logarithm.

$e^{5 lim_{x \to 8} x \cdot \ln (lim_{x \to 8} \frac{8 x}{8 x + 3})}$

Step 3.5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$e^{5 lim_{x \to 8} x \cdot \ln (8 lim_{x \to 8} \frac{x}{8 x + 3})}$

Step 3.6

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

$e^{5 lim_{x \to 8} x \cdot \ln (8 \frac{lim_{x \to 8} x}{lim_{x \to 8} 8 x + 3})}$

Step 3.7

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

$e^{5 lim_{x \to 8} x \cdot \ln (8 \frac{lim_{x \to 8} x}{lim_{x \to 8} 8 x + lim_{x \to 8} 3})}$

Step 3.8

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$e^{5 lim_{x \to 8} x \cdot \ln (8 \frac{lim_{x \to 8} x}{8 lim_{x \to 8} x + lim_{x \to 8} 3})}$

Step 3.9

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$e^{5 lim_{x \to 8} x \cdot \ln (8 \frac{lim_{x \to 8} x}{8 lim_{x \to 8} x + 3})}$

Step 4

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

Tap for more steps...

Step 4.1

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

$e^{5 \cdot 8 \cdot \ln (8 \frac{lim_{x \to 8} x}{8 lim_{x \to 8} x + 3})}$

Step 4.2

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

$e^{5 \cdot 8 \cdot \ln (8 \frac{8}{8 lim_{x \to 8} x + 3})}$

Step 4.3

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

$e^{5 \cdot 8 \cdot \ln (8 \frac{8}{8 \cdot 8 + 3})}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Multiply $5$ by $8$ .

$e^{40 \cdot \ln (8 \frac{8}{8 \cdot 8 + 3})}$

Step 5.2

Simplify $40 \cdot \ln (8 \frac{8}{8 \cdot 8 + 3})$ by moving $40$ inside the logarithm.

$e^{\ln ({(8 \frac{8}{8 \cdot 8 + 3})}^{40})}$

Step 5.3

Exponentiation and log are inverse functions.

${(8 \frac{8}{8 \cdot 8 + 3})}^{40}$

Step 5.4

Simplify the denominator.

Tap for more steps...

Step 5.4.1

Multiply $8$ by $8$ .

${(8 \frac{8}{64 + 3})}^{40}$

Step 5.4.2

Add $64$ and $3$ .

${(8 (\frac{8}{67}))}^{40}$

Step 5.5

Multiply $8 (\frac{8}{67})$ .

Tap for more steps...

Step 5.5.1

Combine $8$ and $\frac{8}{67}$ .

${(\frac{8 \cdot 8}{67})}^{40}$

Step 5.5.2

Multiply $8$ by $8$ .

${(\frac{64}{67})}^{40}$

Step 5.6

Apply the product rule to $\frac{64}{67}$ .

$\frac{64^{40}}{67^{40}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{64^{40}}{67^{40}}$

Decimal Form:

$0.16003200 \dots$