Calculus Examples

$\frac{lim_{y \to 8} 2 - y}{\sqrt{7 + 6 y^{2}}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $8$ .

$\frac{lim_{y \to 8} 2 - lim_{y \to 8} y}{\sqrt{7 + 6 y^{2}}}$

Step 2

Evaluate the limit of $2$ which is constant as $y$ approaches $8$ .

$\frac{2 - lim_{y \to 8} y}{\sqrt{7 + 6 y^{2}}}$

Step 3

Simplify terms.

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Step 3.1

Evaluate the limit of $y$ by plugging in $8$ for $y$ .

$\frac{2 - 8}{\sqrt{7 + 6 y^{2}}}$

Step 3.2

Simplify the answer.

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Step 3.2.1

Subtract $8$ from $2$ .

$\frac{- 6}{\sqrt{7 + 6 y^{2}}}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{6}{\sqrt{7 + 6 y^{2}}}$

Step 3.2.3

Multiply $\frac{6}{\sqrt{7 + 6 y^{2}}}$ by $\frac{\sqrt{7 + 6 y^{2}}}{\sqrt{7 + 6 y^{2}}}$ .

$- (\frac{6}{\sqrt{7 + 6 y^{2}}} \cdot \frac{\sqrt{7 + 6 y^{2}}}{\sqrt{7 + 6 y^{2}}})$

Step 3.2.4

Combine and simplify the denominator.

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Step 3.2.4.1

Multiply $\frac{6}{\sqrt{7 + 6 y^{2}}}$ by $\frac{\sqrt{7 + 6 y^{2}}}{\sqrt{7 + 6 y^{2}}}$ .

$- \frac{6 \sqrt{7 + 6 y^{2}}}{\sqrt{7 + 6 y^{2}} \sqrt{7 + 6 y^{2}}}$

Step 3.2.4.2

Raise $\sqrt{7 + 6 y^{2}}$ to the power of $1$ .

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{\sqrt{7 + 6 y^{2}}}^{1} \sqrt{7 + 6 y^{2}}}$

Step 3.2.4.3

Raise $\sqrt{7 + 6 y^{2}}$ to the power of $1$ .

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{\sqrt{7 + 6 y^{2}}}^{1} {\sqrt{7 + 6 y^{2}}}^{1}}$

Step 3.2.4.4

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

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{\sqrt{7 + 6 y^{2}}}^{1 + 1}}$

Step 3.2.4.5

Add $1$ and $1$ .

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{\sqrt{7 + 6 y^{2}}}^{2}}$

Step 3.2.4.6

Rewrite ${\sqrt{7 + 6 y^{2}}}^{2}$ as $7 + 6 y^{2}$ .

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Step 3.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + 6 y^{2}}$ as ${(7 + 6 y^{2})}^{\frac{1}{2}}$ .

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{({(7 + 6 y^{2})}^{\frac{1}{2}})}^{2}}$

Step 3.2.4.6.2

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

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{(7 + 6 y^{2})}^{\frac{1}{2} \cdot 2}}$

Step 3.2.4.6.3

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

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{(7 + 6 y^{2})}^{\frac{2}{2}}}$

Step 3.2.4.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.6.4.1

Cancel the common factor.

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{(7 + 6 y^{2})}^{\frac{2}{2}}}$

Step 3.2.4.6.4.2

Rewrite the expression.

$- \frac{6 \sqrt{7 + 6 y^{2}}}{{(7 + 6 y^{2})}^{1}}$

Step 3.2.4.6.5

Simplify.

$- \frac{6 \sqrt{7 + 6 y^{2}}}{7 + 6 y^{2}}$