Calculus Examples

$lim_{x \to 8} x \ln (1 + \frac{2}{x})$

Step 1

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

$lim_{x \to 8} x \cdot lim_{x \to 8} \ln (1 + \frac{2}{x})$

Step 2

Move the limit inside the logarithm.

$lim_{x \to 8} x \cdot \ln (lim_{x \to 8} 1 + \frac{2}{x})$

Step 3

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

$lim_{x \to 8} x \cdot \ln (lim_{x \to 8} 1 + lim_{x \to 8} \frac{2}{x})$

Step 4

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

$lim_{x \to 8} x \cdot \ln (1 + lim_{x \to 8} \frac{2}{x})$

Step 5

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

$lim_{x \to 8} x \cdot \ln (1 + 2 lim_{x \to 8} \frac{1}{x})$

Step 6

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

$lim_{x \to 8} x \cdot \ln (1 + 2 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 7

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

$lim_{x \to 8} x \cdot \ln (1 + 2 \frac{1}{lim_{x \to 8} x})$

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 $x$ .

$8 \cdot \ln (1 + 2 \frac{1}{lim_{x \to 8} x})$

Step 8.2

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

$8 \cdot \ln (1 + 2 (\frac{1}{8}))$

Step 9

Simplify the answer.

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Step 9.1

Cancel the common factor of $2$ .

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Step 9.1.1

Factor $2$ out of $8$ .

$8 \cdot \ln (1 + 2 \frac{1}{2 (4)})$

Step 9.1.2

Cancel the common factor.

$8 \cdot \ln (1 + 2 \frac{1}{2 \cdot 4})$

Step 9.1.3

Rewrite the expression.

$8 \cdot \ln (1 + \frac{1}{4})$

Step 9.2

Write $1$ as a fraction with a common denominator.

$8 \cdot \ln (\frac{4}{4} + \frac{1}{4})$

Step 9.3

Combine the numerators over the common denominator.

$8 \cdot \ln (\frac{4 + 1}{4})$

Step 9.4

Add $4$ and $1$ .

$8 \ln (\frac{5}{4})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$8 \ln (\frac{5}{4})$

Decimal Form:

$1.78514841 \dots$