Calculus Examples

$lim_{x \to 4} \sqrt{\frac{7 x^{2}}{6 + x}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 4} \frac{7 x^{2}}{6 + x}}$

Step 2

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

$\sqrt{7 lim_{x \to 4} \frac{x^{2}}{6 + x}}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\sqrt{7 \frac{lim_{x \to 4} x^{2}}{lim_{x \to 4} 6 + x}}$

Step 4

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

$\sqrt{7 \frac{{(lim_{x \to 4} x)}^{2}}{lim_{x \to 4} 6 + x}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\sqrt{7 \frac{{(lim_{x \to 4} x)}^{2}}{lim_{x \to 4} 6 + lim_{x \to 4} x}}$

Step 6

Evaluate the limit of $6$ which is constant as $x$ approaches $4$ .

$\sqrt{7 \frac{{(lim_{x \to 4} x)}^{2}}{6 + lim_{x \to 4} x}}$

Step 7

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\sqrt{7 \frac{4^{2}}{6 + lim_{x \to 4} x}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\sqrt{7 \frac{4^{2}}{6 + 4}}$

Step 8

Simplify the answer.

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Step 8.1

Combine $7$ and $\frac{4^{2}}{6 + 4}$ .

$\sqrt{\frac{7 \cdot 4^{2}}{6 + 4}}$

Step 8.2

Add $6$ and $4$ .

$\sqrt{\frac{7 \cdot 4^{2}}{10}}$

Step 8.3

Raise $4$ to the power of $2$ .

$\sqrt{\frac{7 \cdot 16}{10}}$

Step 8.4

Multiply $7$ by $16$ .

$\sqrt{\frac{112}{10}}$

Step 8.5

Cancel the common factor of $112$ and $10$ .

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Step 8.5.1

Factor $2$ out of $112$ .

$\sqrt{\frac{2 (56)}{10}}$

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Factor $2$ out of $10$ .

$\sqrt{\frac{2 \cdot 56}{2 \cdot 5}}$

Step 8.5.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 56}{2 \cdot 5}}$

Step 8.5.2.3

Rewrite the expression.

$\sqrt{\frac{56}{5}}$

Step 8.6

Rewrite $\sqrt{\frac{56}{5}}$ as $\frac{\sqrt{56}}{\sqrt{5}}$ .

$\frac{\sqrt{56}}{\sqrt{5}}$

Step 8.7

Simplify the numerator.

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Step 8.7.1

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 8.7.1.1

Factor $4$ out of $56$ .

$\frac{\sqrt{4 (14)}}{\sqrt{5}}$

Step 8.7.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 14}}{\sqrt{5}}$

Step 8.7.2

Pull terms out from under the radical.

$\frac{2 \sqrt{14}}{\sqrt{5}}$

Step 8.8

Multiply $\frac{2 \sqrt{14}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{14}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8.9

Combine and simplify the denominator.

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Step 8.9.1

Multiply $\frac{2 \sqrt{14}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{14} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.9.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{14} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 8.9.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{14} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 8.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{14} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 8.9.5

Add $1$ and $1$ .

$\frac{2 \sqrt{14} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 8.9.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 \sqrt{14} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 8.9.6.2

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

$\frac{2 \sqrt{14} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 8.9.6.3

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

$\frac{2 \sqrt{14} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.9.6.4

Cancel the common factor of $2$ .

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Step 8.9.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{14} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.9.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{14} \sqrt{5}}{5^{1}}$

Step 8.9.6.5

Evaluate the exponent.

$\frac{2 \sqrt{14} \sqrt{5}}{5}$

Step 8.10

Simplify the numerator.

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Step 8.10.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{5 \cdot 14}}{5}$

Step 8.10.2

Multiply $5$ by $14$ .

$\frac{2 \sqrt{70}}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{70}}{5}$

Decimal Form:

$3.34664010 \dots$