Calculus Examples

$lim_{x \to 2} 3^{x^{2} - 3}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit into the exponent.

$3^{lim_{x \to 2} x^{2} - 3}$

Step 1.2

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

$3^{lim_{x \to 2} x^{2} - lim_{x \to 2} 3}$

Step 1.3

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

$3^{{(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 3}$

Step 1.4

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

$3^{{(lim_{x \to 2} x)}^{2} - 1 \cdot 3}$

Step 2

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

$3^{2^{2} - 1 \cdot 3}$

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 $2$ .

$3^{4 - 1 \cdot 3}$

Step 3.1.2

Multiply $- 1$ by $3$ .

$3^{4 - 3}$

Step 3.2

Subtract $3$ from $4$ .

$3^{1}$

Step 3.3

Evaluate the exponent.

$3$