Calculus Examples

$lim_{x \to 8} \frac{\ln (2 x)}{\ln (3 x)}$

Step 1

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

$\frac{lim_{x \to 8} \ln (2 x)}{lim_{x \to 8} \ln (3 x)}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} 2 x)}{lim_{x \to 8} \ln (3 x)}$

Step 3

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

$\frac{\ln (2 lim_{x \to 8} x)}{lim_{x \to 8} \ln (3 x)}$

Step 4

Move the limit inside the logarithm.

$\frac{\ln (2 lim_{x \to 8} x)}{\ln (lim_{x \to 8} 3 x)}$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (2 lim_{x \to 8} x)}{\ln (3 lim_{x \to 8} x)}$

Step 6

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

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Step 6.1

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

$\frac{\ln (2 \cdot 8)}{\ln (3 lim_{x \to 8} x)}$

Step 6.2

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

$\frac{\ln (2 \cdot 8)}{\ln (3 \cdot 8)}$

Step 7

Simplify the answer.

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Step 7.1

Multiply $2$ by $8$ .

$\frac{\ln (16)}{\ln (3 \cdot 8)}$

Step 7.2

Multiply $3$ by $8$ .

$\frac{\ln (16)}{\ln (24)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (16)}{\ln (24)}$

Decimal Form:

$0.87241716 \dots$