Calculus Examples

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

Step 1

Divide the numerator and denominator by the highest power of $y$ in the denominator, which is $y = - \sqrt{y^{2}}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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

Step 2.2

Cancel the common factor of $y^{2}$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $6$ by $1$ .

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

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $- \infty$ .

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

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- \infty$ .

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

Step 2.5

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

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

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{y}$ approaches $0$ .

$\frac{2 \cdot 0 - lim_{y \to - \infty} 1}{lim_{y \to - \infty} - \sqrt{\frac{7}{y^{2}} + 6}}$

Step 4

Evaluate the limit.

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Step 4.1

Evaluate the limit of $1$ which is constant as $y$ approaches $- \infty$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{lim_{y \to - \infty} - \sqrt{\frac{7}{y^{2}} + 6}}$

Step 4.2

Move the term $- 1$ outside of the limit because it is constant with respect to $y$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{- lim_{y \to - \infty} \sqrt{\frac{7}{y^{2}} + 6}}$

Step 4.3

Move the limit under the radical sign.

$\frac{2 \cdot 0 - 1 \cdot 1}{- \sqrt{lim_{y \to - \infty} \frac{7}{y^{2}} + 6}}$

Step 4.4

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- \infty$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{- \sqrt{lim_{y \to - \infty} \frac{7}{y^{2}} + lim_{y \to - \infty} 6}}$

Step 4.5

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

$\frac{2 \cdot 0 - 1 \cdot 1}{- \sqrt{7 lim_{y \to - \infty} \frac{1}{y^{2}} + lim_{y \to - \infty} 6}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{y^{2}}$ approaches $0$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{- \sqrt{7 \cdot 0 + lim_{y \to - \infty} 6}}$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit of $6$ which is constant as $y$ approaches $- \infty$ .

$\frac{2 \cdot 0 - 1 \cdot 1}{- \sqrt{7 \cdot 0 + 6}}$

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $2$ by $0$ .

$\frac{0 - 1 \cdot 1}{- \sqrt{7 \cdot 0 + 6}}$

Step 6.2.1.2

Multiply $- 1$ by $1$ .

$\frac{0 - 1}{- \sqrt{7 \cdot 0 + 6}}$

Step 6.2.1.3

Subtract $1$ from $0$ .

$\frac{- 1}{- \sqrt{7 \cdot 0 + 6}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $7$ by $0$ .

$\frac{- 1}{- \sqrt{0 + 6}}$

Step 6.2.2.2

Add $0$ and $6$ .

$\frac{- 1}{- \sqrt{6}}$

Step 6.2.3

Dividing two negative values results in a positive value.

$\frac{1}{\sqrt{6}}$

Step 6.2.4

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 6.2.5

Combine and simplify the denominator.

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Step 6.2.5.1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 6.2.5.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 6.2.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 6.2.5.4

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

$\frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 6.2.5.5

Add $1$ and $1$ .

$\frac{\sqrt{6}}{{\sqrt{6}}^{2}}$

Step 6.2.5.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 6.2.5.6.1

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

$\frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 6.2.5.6.2

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

$\frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 6.2.5.6.3

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

$\frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Step 6.2.5.6.4

Cancel the common factor of $2$ .

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Step 6.2.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{6}}{6^{\frac{2}{2}}}$

Step 6.2.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{6}}{6^{1}}$

Step 6.2.5.6.5

Evaluate the exponent.

$\frac{\sqrt{6}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{6}$

Decimal Form:

$0.40824829 \dots$