Calculus Examples

$lim_{x \to 10} \sqrt{\frac{10 x}{2 x + 5}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 10} \frac{10 x}{2 x + 5}}$

Step 2

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{10 lim_{x \to 10} \frac{x}{2 x + 5}}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $10$ .

$\sqrt{10 \frac{lim_{x \to 10} x}{lim_{x \to 10} 2 x + 5}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $10$ .

$\sqrt{10 \frac{lim_{x \to 10} x}{lim_{x \to 10} 2 x + lim_{x \to 10} 5}}$

Step 5

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

$\sqrt{10 \frac{lim_{x \to 10} x}{2 lim_{x \to 10} x + lim_{x \to 10} 5}}$

Step 6

Evaluate the limit of $5$ which is constant as $x$ approaches $10$ .

$\sqrt{10 \frac{lim_{x \to 10} x}{2 lim_{x \to 10} x + 5}}$

Step 7

Evaluate the limits by plugging in $10$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $10$ for $x$ .

$\sqrt{10 \frac{10}{2 lim_{x \to 10} x + 5}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $10$ for $x$ .

$\sqrt{10 \frac{10}{2 \cdot 10 + 5}}$

Step 8

Simplify the answer.

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Step 8.1

Combine $10$ and $\frac{10}{2 \cdot 10 + 5}$ .

$\sqrt{\frac{10 \cdot 10}{2 \cdot 10 + 5}}$

Step 8.2

Multiply $2$ by $10$ .

$\sqrt{\frac{10 \cdot 10}{20 + 5}}$

Step 8.3

Add $20$ and $5$ .

$\sqrt{\frac{10 \cdot 10}{25}}$

Step 8.4

Reduce the expression $\frac{10 \cdot 10}{25}$ by cancelling the common factors.

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Step 8.4.1

Factor $5$ out of $10 \cdot 10$ .

$\sqrt{\frac{5 (2 \cdot 10)}{25}}$

Step 8.4.2

Factor $5$ out of $25$ .

$\sqrt{\frac{5 (2 \cdot 10)}{5 (5)}}$

Step 8.4.3

Cancel the common factor.

$\sqrt{\frac{5 (2 \cdot 10)}{5 \cdot 5}}$

Step 8.4.4

Rewrite the expression.

$\sqrt{\frac{2 \cdot 10}{5}}$

Step 8.5

Cancel the common factor of $10$ and $5$ .

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Step 8.5.1

Factor $5$ out of $2 \cdot 10$ .

$\sqrt{\frac{5 (2 \cdot 2)}{5}}$

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Factor $5$ out of $5$ .

$\sqrt{\frac{5 (2 \cdot 2)}{5 (1)}}$

Step 8.5.2.2

Cancel the common factor.

$\sqrt{\frac{5 (2 \cdot 2)}{5 \cdot 1}}$

Step 8.5.2.3

Rewrite the expression.

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

Step 8.5.2.4

Divide $2 \cdot 2$ by $1$ .

$\sqrt{2 \cdot 2}$

Step 8.6

Multiply $2$ by $2$ .

$\sqrt{4}$

Step 8.7

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

$\sqrt{2^{2}}$

Step 8.8

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

$2$