Calculus Examples

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

Step 1

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

$lim_{x \to \sqrt{2}} x^{2} + 3 \cdot lim_{x \to \sqrt{2}} x^{2} - 1$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $\sqrt{2}$ .

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

Step 8

Evaluate the limits by plugging in $\sqrt{2}$ for all occurrences of $x$ .

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Step 8.1

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

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

Step 8.2

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

$({\sqrt{2}}^{2} + 3) \cdot ({\sqrt{2}}^{2} - 1 \cdot 1)$

Step 9

Simplify the answer.

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.1.4

Cancel the common factor of $2$ .

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Step 9.1.4.1

Cancel the common factor.

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

Step 9.1.4.2

Rewrite the expression.

$(2^{1} + 3) \cdot ({\sqrt{2}}^{2} - 1 \cdot 1)$

Step 9.1.5

Evaluate the exponent.

$(2 + 3) \cdot ({\sqrt{2}}^{2} - 1 \cdot 1)$

Step 9.2

Add $2$ and $3$ .

$5 \cdot ({\sqrt{2}}^{2} - 1 \cdot 1)$

Step 9.3

Simplify each term.

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Step 9.3.1

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

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Step 9.3.1.1

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

$5 \cdot ({(2^{\frac{1}{2}})}^{2} - 1 \cdot 1)$

Step 9.3.1.2

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

$5 \cdot (2^{\frac{1}{2} \cdot 2} - 1 \cdot 1)$

Step 9.3.1.3

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

$5 \cdot (2^{\frac{2}{2}} - 1 \cdot 1)$

Step 9.3.1.4

Cancel the common factor of $2$ .

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Step 9.3.1.4.1

Cancel the common factor.

$5 \cdot (2^{\frac{2}{2}} - 1 \cdot 1)$

Step 9.3.1.4.2

Rewrite the expression.

$5 \cdot (2^{1} - 1 \cdot 1)$

Step 9.3.1.5

Evaluate the exponent.

$5 \cdot (2 - 1 \cdot 1)$

Step 9.3.2

Multiply $- 1$ by $1$ .

$5 \cdot (2 - 1)$

Step 9.4

Subtract $1$ from $2$ .

$5 \cdot 1$

Step 9.5

Multiply $5$ by $1$ .

$5$