Calculus Examples

$lim_{x \to 8} \frac{5 x}{x - 1} - \frac{5}{\ln (x)}$

Step 1

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

$lim_{x \to 8} \frac{5 x}{x - 1} - lim_{x \to 8} \frac{5}{\ln (x)}$

Step 2

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

$5 lim_{x \to 8} \frac{x}{x - 1} - lim_{x \to 8} \frac{5}{\ln (x)}$

Step 3

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

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1} - lim_{x \to 8} \frac{5}{\ln (x)}$

Step 4

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

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - lim_{x \to 8} 1} - lim_{x \to 8} \frac{5}{\ln (x)}$

Step 5

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

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1} - lim_{x \to 8} \frac{5}{\ln (x)}$

Step 6

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

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1} - 5 lim_{x \to 8} \frac{1}{\ln (x)}$

Step 7

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

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1} - 5 \frac{lim_{x \to 8} 1}{lim_{x \to 8} \ln (x)}$

Step 8

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

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1} - 5 \frac{1}{lim_{x \to 8} \ln (x)}$

Step 9

Move the limit inside the logarithm.

$5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1} - 5 \frac{1}{\ln (lim_{x \to 8} x)}$

Step 10

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

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Step 10.1

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

$5 \frac{8}{lim_{x \to 8} x - 1 \cdot 1} - 5 \frac{1}{\ln (lim_{x \to 8} x)}$

Step 10.2

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

$5 \frac{8}{8 - 1 \cdot 1} - 5 \frac{1}{\ln (lim_{x \to 8} x)}$

Step 10.3

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

$5 \frac{8}{8 - 1 \cdot 1} - 5 \frac{1}{\ln (8)}$

Step 11

Simplify each term.

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Step 11.1

Simplify the denominator.

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Step 11.1.1

Multiply $- 1$ by $1$ .

$5 \frac{8}{8 - 1} - 5 \frac{1}{\ln (8)}$

Step 11.1.2

Subtract $1$ from $8$ .

$5 (\frac{8}{7}) - 5 \frac{1}{\ln (8)}$

Step 11.2

Multiply $5 (\frac{8}{7})$ .

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Step 11.2.1

Combine $5$ and $\frac{8}{7}$ .

$\frac{5 \cdot 8}{7} - 5 \frac{1}{\ln (8)}$

Step 11.2.2

Multiply $5$ by $8$ .

$\frac{40}{7} - 5 \frac{1}{\ln (8)}$

Step 11.3

Combine $- 5$ and $\frac{1}{\ln (8)}$ .

$\frac{40}{7} + \frac{- 5}{\ln (8)}$

Step 11.4

Move the negative in front of the fraction.

$\frac{40}{7} - \frac{5}{\ln (8)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{40}{7} - \frac{5}{\ln (8)}$

Decimal Form:

$3.30979397 \dots$