Calculus Examples

$lim_{x \to - 0.2} \sqrt{\frac{x + 5}{x + 1}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to - 0.2} \frac{x + 5}{x + 1}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 0.2$ .

$\sqrt{\frac{lim_{x \to - 0.2} x + 5}{lim_{x \to - 0.2} x + 1}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 0.2$ .

$\sqrt{\frac{lim_{x \to - 0.2} x + lim_{x \to - 0.2} 5}{lim_{x \to - 0.2} x + 1}}$

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $- 0.2$ .

$\sqrt{\frac{lim_{x \to - 0.2} x + 5}{lim_{x \to - 0.2} x + 1}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 0.2$ .

$\sqrt{\frac{lim_{x \to - 0.2} x + 5}{lim_{x \to - 0.2} x + lim_{x \to - 0.2} 1}}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $- 0.2$ .

$\sqrt{\frac{lim_{x \to - 0.2} x + 5}{lim_{x \to - 0.2} x + 1}}$

Step 7

Evaluate the limits by plugging in $- 0.2$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 0.2$ for $x$ .

$\sqrt{\frac{- 0.2 + 5}{lim_{x \to - 0.2} x + 1}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 0.2$ for $x$ .

$\sqrt{\frac{- 0.2 + 5}{- 0.2 + 1}}$

Step 8

Simplify the answer.

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Step 8.1

Add $- 0.2$ and $5$ .

$\sqrt{\frac{4.8}{- 0.2 + 1}}$

Step 8.2

Add $- 0.2$ and $1$ .

$\sqrt{\frac{4.8}{0.8}}$

Step 8.3

Divide $4.8$ by $0.8$ .

$\sqrt{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\sqrt{6}$

Decimal Form:

$2.44948974 \dots$