Evaluate limit as x approaches 8 of (5x)/(x-1)-5/( natural log of x)

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

Take the limit of each term.

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

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

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

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

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

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

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

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

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

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

Move the limit inside the logarithm.

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

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

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

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

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

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

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

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

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

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

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

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

Simplify the answer.

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Multiply $5$ by $8$ .

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

Subtract $1$ from $8$ .

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.30979397 \dots$

Evaluate limit as x approaches 8 of (5x)/(x-1)-5/( natural log of x)

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