Calculus Examples

$lim_{x \to \sqrt{2}} (2 x^{2} + 1) (7 x^{2} + 13)$

Step 1

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

$lim_{x \to \sqrt{2}} 2 x^{2} + 1 \cdot lim_{x \to \sqrt{2}} 7 x^{2} + 13$

Step 2

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

$(lim_{x \to \sqrt{2}} 2 x^{2} + lim_{x \to \sqrt{2}} 1) \cdot lim_{x \to \sqrt{2}} 7 x^{2} + 13$

Step 3

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

$(2 lim_{x \to \sqrt{2}} x^{2} + lim_{x \to \sqrt{2}} 1) \cdot lim_{x \to \sqrt{2}} 7 x^{2} + 13$

Step 4

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

$(2 {(lim_{x \to \sqrt{2}} x)}^{2} + lim_{x \to \sqrt{2}} 1) \cdot lim_{x \to \sqrt{2}} 7 x^{2} + 13$

Step 5

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

$(2 {(lim_{x \to \sqrt{2}} x)}^{2} + 1) \cdot lim_{x \to \sqrt{2}} 7 x^{2} + 13$

Step 6

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

$(2 {(lim_{x \to \sqrt{2}} x)}^{2} + 1) \cdot (lim_{x \to \sqrt{2}} 7 x^{2} + lim_{x \to \sqrt{2}} 13)$

Step 7

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$(2 {(lim_{x \to \sqrt{2}} x)}^{2} + 1) \cdot (7 lim_{x \to \sqrt{2}} x^{2} + lim_{x \to \sqrt{2}} 13)$

Step 8

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

$(2 {(lim_{x \to \sqrt{2}} x)}^{2} + 1) \cdot (7 {(lim_{x \to \sqrt{2}} x)}^{2} + lim_{x \to \sqrt{2}} 13)$

Step 9

Evaluate the limit of $13$ which is constant as $x$ approaches $\sqrt{2}$ .

$(2 {(lim_{x \to \sqrt{2}} x)}^{2} + 1) \cdot (7 {(lim_{x \to \sqrt{2}} x)}^{2} + 13)$

Step 10

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

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Step 10.1

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

$(2 {\sqrt{2}}^{2} + 1) \cdot (7 {(lim_{x \to \sqrt{2}} x)}^{2} + 13)$

Step 10.2

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

$(2 {\sqrt{2}}^{2} + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

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Step 11.1.1.1

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

$(2 {(2^{\frac{1}{2}})}^{2} + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.1.1.2

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

$(2 \cdot 2^{\frac{1}{2} \cdot 2} + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.1.1.3

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

$(2 \cdot 2^{\frac{2}{2}} + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.1.1.4

Cancel the common factor of $2$ .

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Step 11.1.1.4.1

Cancel the common factor.

$(2 \cdot 2^{\frac{2}{2}} + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.1.1.4.2

Rewrite the expression.

$(2 \cdot 2^{1} + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.1.1.5

Evaluate the exponent.

$(2 \cdot 2 + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.1.2

Multiply $2$ by $2$ .

$(4 + 1) \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.2

Add $4$ and $1$ .

$5 \cdot (7 {\sqrt{2}}^{2} + 13)$

Step 11.3

Simplify each term.

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Step 11.3.1

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

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Step 11.3.1.1

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

$5 \cdot (7 {(2^{\frac{1}{2}})}^{2} + 13)$

Step 11.3.1.2

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

$5 \cdot (7 \cdot 2^{\frac{1}{2} \cdot 2} + 13)$

Step 11.3.1.3

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

$5 \cdot (7 \cdot 2^{\frac{2}{2}} + 13)$

Step 11.3.1.4

Cancel the common factor of $2$ .

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Step 11.3.1.4.1

Cancel the common factor.

$5 \cdot (7 \cdot 2^{\frac{2}{2}} + 13)$

Step 11.3.1.4.2

Rewrite the expression.

$5 \cdot (7 \cdot 2^{1} + 13)$

Step 11.3.1.5

Evaluate the exponent.

$5 \cdot (7 \cdot 2 + 13)$

Step 11.3.2

Multiply $7$ by $2$ .

$5 \cdot (14 + 13)$

Step 11.4

Add $14$ and $13$ .

$5 \cdot 27$

Step 11.5

Multiply $5$ by $27$ .

$135$