Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\sqrt{2}$ .

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

Step 1.2

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

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

Step 1.3

Evaluate the limit of $3$ which is constant as $x$ approaches $\sqrt{2}$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $\sqrt{2}$ for $x$ .

${\sqrt{2}}^{2} + 3$

Step 3

Simplify the answer.

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Step 3.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{\frac{1}{2} \cdot 2} + 3$

Step 3.1.3

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

$2^{\frac{2}{2}} + 3$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Cancel the common factor.

$2^{\frac{2}{2}} + 3$

Step 3.1.4.2

Rewrite the expression.

$2^{1} + 3$

Step 3.1.5

Evaluate the exponent.

$2 + 3$

Step 3.2

Add $2$ and $3$ .

$5$