Calculus Examples

$lim_{x \to 8} \frac{\ln (5 x + 7)}{\ln (6 x + 10) + 1}$

Step 1

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

$\frac{lim_{x \to 8} \ln (5 x + 7)}{lim_{x \to 8} \ln (6 x + 10) + 1}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} 5 x + 7)}{lim_{x \to 8} \ln (6 x + 10) + 1}$

Step 3

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

$\frac{\ln (lim_{x \to 8} 5 x + lim_{x \to 8} 7)}{lim_{x \to 8} \ln (6 x + 10) + 1}$

Step 4

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

$\frac{\ln (5 lim_{x \to 8} x + lim_{x \to 8} 7)}{lim_{x \to 8} \ln (6 x + 10) + 1}$

Step 5

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{\ln (5 lim_{x \to 8} x + 7)}{lim_{x \to 8} \ln (6 x + 10) + 1}$

Step 6

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

$\frac{\ln (5 lim_{x \to 8} x + 7)}{lim_{x \to 8} \ln (6 x + 10) + lim_{x \to 8} 1}$

Step 7

Move the limit inside the logarithm.

$\frac{\ln (5 lim_{x \to 8} x + 7)}{\ln (lim_{x \to 8} 6 x + 10) + lim_{x \to 8} 1}$

Step 8

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

$\frac{\ln (5 lim_{x \to 8} x + 7)}{\ln (lim_{x \to 8} 6 x + lim_{x \to 8} 10) + lim_{x \to 8} 1}$

Step 9

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (5 lim_{x \to 8} x + 7)}{\ln (6 lim_{x \to 8} x + lim_{x \to 8} 10) + lim_{x \to 8} 1}$

Step 10

Evaluate the limit of $10$ which is constant as $x$ approaches $8$ .

$\frac{\ln (5 lim_{x \to 8} x + 7)}{\ln (6 lim_{x \to 8} x + 10) + lim_{x \to 8} 1}$

Step 11

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

$\frac{\ln (5 lim_{x \to 8} x + 7)}{\ln (6 lim_{x \to 8} x + 10) + 1}$

Step 12

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

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Step 12.1

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

$\frac{\ln (5 \cdot 8 + 7)}{\ln (6 lim_{x \to 8} x + 10) + 1}$

Step 12.2

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

$\frac{\ln (5 \cdot 8 + 7)}{\ln (6 \cdot 8 + 10) + 1}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

Multiply $5$ by $8$ .

$\frac{\ln (40 + 7)}{\ln (6 \cdot 8 + 10) + 1}$

Step 13.1.2

Add $40$ and $7$ .

$\frac{\ln (47)}{\ln (6 \cdot 8 + 10) + 1}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $6$ by $8$ .

$\frac{\ln (47)}{\ln (48 + 10) + 1}$

Step 13.2.2

Add $48$ and $10$ .

$\frac{\ln (47)}{\ln (58) + 1}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (47)}{\ln (58) + 1}$

Decimal Form:

$0.76083212 \dots$