Calculus Examples

$lim_{x \to 8} \frac{\ln ({(x + 2)}^{2})}{\ln (x^{4})}$

Step 1

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

$\frac{lim_{x \to 8} \ln ({(x + 2)}^{2})}{lim_{x \to 8} \ln (x^{4})}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} {(x + 2)}^{2})}{lim_{x \to 8} \ln (x^{4})}$

Step 3

Move the exponent $2$ from ${(x + 2)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{\ln ({(lim_{x \to 8} x + 2)}^{2})}{lim_{x \to 8} \ln (x^{4})}$

Step 4

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

$\frac{\ln ({(lim_{x \to 8} x + lim_{x \to 8} 2)}^{2})}{lim_{x \to 8} \ln (x^{4})}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{\ln ({(lim_{x \to 8} x + 2)}^{2})}{lim_{x \to 8} \ln (x^{4})}$

Step 6

Move the limit inside the logarithm.

$\frac{\ln ({(lim_{x \to 8} x + 2)}^{2})}{\ln (lim_{x \to 8} x^{4})}$

Step 7

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{\ln ({(lim_{x \to 8} x + 2)}^{2})}{\ln ({(lim_{x \to 8} x)}^{4})}$

Step 8

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

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Step 8.1

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

$\frac{\ln ({(8 + 2)}^{2})}{\ln ({(lim_{x \to 8} x)}^{4})}$

Step 8.2

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

$\frac{\ln ({(8 + 2)}^{2})}{\ln (8^{4})}$

Step 9

Simplify the answer.

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Step 9.1

Expand $\ln ({(8 + 2)}^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \ln (8 + 2)}{\ln (8^{4})}$

Step 9.2

Expand $\ln (8^{4})$ by moving $4$ outside the logarithm.

$\frac{2 \ln (8 + 2)}{4 \ln (8)}$

Step 9.3

Cancel the common factor of $2$ and $4$ .

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Step 9.3.1

Factor $2$ out of $2 \ln (8 + 2)$ .

$\frac{2 (\ln (8 + 2))}{4 \ln (8)}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $4 \ln (8)$ .

$\frac{2 (\ln (8 + 2))}{2 (2 \ln (8))}$

Step 9.3.2.2

Cancel the common factor.

$\frac{2 \ln (8 + 2)}{2 (2 \ln (8))}$

Step 9.3.2.3

Rewrite the expression.

$\frac{\ln (8 + 2)}{2 \ln (8)}$

Step 9.4

Add $8$ and $2$ .

$\frac{\ln (10)}{2 \ln (8)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (10)}{2 \ln (8)}$

Decimal Form:

$0.55365468 \dots$