Calculus Examples

$lim_{x \to 2} (2) (\frac{x^{2}}{2} + 2)$

Step 1

Evaluate the limit.

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Step 1.1

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

$2 lim_{x \to 2} \frac{x^{2}}{2} + 2$

Step 1.2

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

$2 (lim_{x \to 2} \frac{x^{2}}{2} + lim_{x \to 2} 2)$

Step 1.3

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

$2 (\frac{1}{2} \cdot 2^{2} + 2)$

Step 3

Simplify the answer.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Factor $2$ out of $2^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

$2 (2 + 2)$

Step 3.2

Add $2$ and $2$ .

$2 \cdot 4$

Step 3.3

Multiply $2$ by $4$ .

$8$