Calculus Examples

$\frac{lim_{x \to 11} \sqrt{x^{2} - 2 x - 63}}{34 - 4 x}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 11} x^{2} - 2 x - 63}}{34 - 4 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $11$ .

$\frac{\sqrt{lim_{x \to 11} x^{2} - lim_{x \to 11} 2 x - lim_{x \to 11} 63}}{34 - 4 x}$

Step 3

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

$\frac{\sqrt{{(lim_{x \to 11} x)}^{2} - lim_{x \to 11} 2 x - lim_{x \to 11} 63}}{34 - 4 x}$

Step 4

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

$\frac{\sqrt{{(lim_{x \to 11} x)}^{2} - 2 lim_{x \to 11} x - lim_{x \to 11} 63}}{34 - 4 x}$

Step 5

Evaluate the limit of $63$ which is constant as $x$ approaches $11$ .

$\frac{\sqrt{{(lim_{x \to 11} x)}^{2} - 2 lim_{x \to 11} x - 1 \cdot 63}}{34 - 4 x}$

Step 6

Evaluate the limits by plugging in $11$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $11$ for $x$ .

$\frac{\sqrt{11^{2} - 2 lim_{x \to 11} x - 1 \cdot 63}}{34 - 4 x}$

Step 6.2

Evaluate the limit of $x$ by plugging in $11$ for $x$ .

$\frac{\sqrt{11^{2} - 2 \cdot 11 - 1 \cdot 63}}{34 - 4 x}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $11$ to the power of $2$ .

$\frac{\sqrt{121 - 2 \cdot 11 - 1 \cdot 63}}{34 - 4 x}$

Step 7.1.2

Multiply $- 2$ by $11$ .

$\frac{\sqrt{121 - 22 - 1 \cdot 63}}{34 - 4 x}$

Step 7.1.3

Multiply $- 1$ by $63$ .

$\frac{\sqrt{121 - 22 - 63}}{34 - 4 x}$

Step 7.1.4

Subtract $22$ from $121$ .

$\frac{\sqrt{99 - 63}}{34 - 4 x}$

Step 7.1.5

Subtract $63$ from $99$ .

$\frac{\sqrt{36}}{34 - 4 x}$

Step 7.1.6

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}}}{34 - 4 x}$

Step 7.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\frac{6}{34 - 4 x}$

Step 7.2

Factor $2$ out of $34 - 4 x$ .

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Step 7.2.1

Factor $2$ out of $34$ .

$\frac{6}{2 (17) - 4 x}$

Step 7.2.2

Factor $2$ out of $- 4 x$ .

$\frac{6}{2 (17) + 2 (- 2 x)}$

Step 7.2.3

Factor $2$ out of $2 (17) + 2 (- 2 x)$ .

$\frac{6}{2 (17 - 2 x)}$

Step 7.3

Cancel the common factor of $6$ and $2$ .

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Step 7.3.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2 (17 - 2 x)}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Cancel the common factor.

$\frac{2 \cdot 3}{2 (17 - 2 x)}$

Step 7.3.2.2

Rewrite the expression.

$\frac{3}{17 - 2 x}$