Calculus Examples

$lim_{x \to 2} \frac{3^{x} + x^{2}}{2}$

Step 1

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

$\frac{1}{2} lim_{x \to 2} 3^{x} + x^{2}$

Step 2

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

$\frac{1}{2} (lim_{x \to 2} 3^{x} + lim_{x \to 2} x^{2})$

Step 3

Move the limit into the exponent.

$\frac{1}{2} (3^{lim_{x \to 2} x} + lim_{x \to 2} x^{2})$

Step 4

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

$\frac{1}{2} (3^{lim_{x \to 2} x} + {(lim_{x \to 2} x)}^{2})$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 5.1

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

$\frac{1}{2} (3^{2} + {(lim_{x \to 2} x)}^{2})$

Step 5.2

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

$\frac{1}{2} (3^{2} + 2^{2})$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

Raise $3$ to the power of $2$ .

$\frac{1}{2} (9 + 2^{2})$

Step 6.1.2

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

$\frac{1}{2} (9 + 4)$

Step 6.2

Add $9$ and $4$ .

$\frac{1}{2} \cdot 13$

Step 6.3

Combine $\frac{1}{2}$ and $13$ .

$\frac{13}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{2}$

Decimal Form:

$6.5$