Calculus Examples

$lim_{x \to 2} \ln (x^{2} - 2)$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the logarithm.

$\ln (lim_{x \to 2} x^{2} - 2)$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\ln (lim_{x \to 2} x^{2} - lim_{x \to 2} 2)$

Step 1.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\ln ({(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 2)$

Step 1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

$\ln ({(lim_{x \to 2} x)}^{2} - 1 \cdot 2)$

Step 2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\ln (2^{2} - 1 \cdot 2)$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise $2$ to the power of $2$ .

$\ln (4 - 1 \cdot 2)$

Step 3.1.2

Multiply $- 1$ by $2$ .

$\ln (4 - 2)$

Step 3.2

Subtract $2$ from $4$ .

$\ln (2)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\ln (2)$

Decimal Form:

$0.69314718 \dots$