Calculus Examples

$lim_{x \to 2} \sqrt{\frac{4 x + 10}{2 x - 2}}$

Step 1

Cancel the common factor of $4 x + 10$ and $2 x - 2$ .

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Step 1.1

Factor $2$ out of $4 x$ .

$lim_{x \to 2} \sqrt{\frac{2 (2 x) + 10}{2 x - 2}}$

Step 1.2

Factor $2$ out of $10$ .

$lim_{x \to 2} \sqrt{\frac{2 (2 x) + 2 (5)}{2 x - 2}}$

Step 1.3

Factor $2$ out of $2 (2 x) + 2 (5)$ .

$lim_{x \to 2} \sqrt{\frac{2 (2 x + 5)}{2 x - 2}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $2 x$ .

$lim_{x \to 2} \sqrt{\frac{2 (2 x + 5)}{2 (x) - 2}}$

Step 1.4.2

Factor $2$ out of $- 2$ .

$lim_{x \to 2} \sqrt{\frac{2 (2 x + 5)}{2 (x) + 2 (- 1)}}$

Step 1.4.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$lim_{x \to 2} \sqrt{\frac{2 (2 x + 5)}{2 (x - 1)}}$

Step 1.4.4

Cancel the common factor.

$lim_{x \to 2} \sqrt{\frac{2 (2 x + 5)}{2 (x - 1)}}$

Step 1.4.5

Rewrite the expression.

$lim_{x \to 2} \sqrt{\frac{2 x + 5}{x - 1}}$

Step 2

Move the limit under the radical sign.

$\sqrt{lim_{x \to 2} \frac{2 x + 5}{x - 1}}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\sqrt{\frac{lim_{x \to 2} 2 x + 5}{lim_{x \to 2} x - 1}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\sqrt{\frac{lim_{x \to 2} 2 x + lim_{x \to 2} 5}{lim_{x \to 2} x - 1}}$

Step 5

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

$\sqrt{\frac{2 lim_{x \to 2} x + lim_{x \to 2} 5}{lim_{x \to 2} x - 1}}$

Step 6

Evaluate the limit of $5$ which is constant as $x$ approaches $2$ .

$\sqrt{\frac{2 lim_{x \to 2} x + 5}{lim_{x \to 2} x - 1}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\sqrt{\frac{2 lim_{x \to 2} x + 5}{lim_{x \to 2} x - lim_{x \to 2} 1}}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$\sqrt{\frac{2 lim_{x \to 2} x + 5}{lim_{x \to 2} x - 1 \cdot 1}}$

Step 9

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\sqrt{\frac{2 \cdot 2 + 5}{lim_{x \to 2} x - 1 \cdot 1}}$

Step 9.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\sqrt{\frac{2 \cdot 2 + 5}{2 - 1 \cdot 1}}$

Step 10

Simplify the answer.

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Step 10.1

Multiply $2$ by $2$ .

$\sqrt{\frac{4 + 5}{2 - 1 \cdot 1}}$

Step 10.2

Add $4$ and $5$ .

$\sqrt{\frac{9}{2 - 1 \cdot 1}}$

Step 10.3

Multiply $- 1$ by $1$ .

$\sqrt{\frac{9}{2 - 1}}$

Step 10.4

Subtract $1$ from $2$ .

$\sqrt{\frac{9}{1}}$

Step 10.5

Divide $9$ by $1$ .

$\sqrt{9}$

Step 10.6

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 10.7

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

$3$