Calculus Examples

$lim_{x \to - 0.25} - \sqrt{\frac{x + 4}{x + 1}}$

Step 1

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

$- lim_{x \to - 0.25} \sqrt{\frac{x + 4}{x + 1}}$

Step 2

Move the limit under the radical sign.

$- \sqrt{lim_{x \to - 0.25} \frac{x + 4}{x + 1}}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 0.25$ .

$- \sqrt{\frac{lim_{x \to - 0.25} x + 4}{lim_{x \to - 0.25} x + 1}}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 0.25$ .

$- \sqrt{\frac{lim_{x \to - 0.25} x + lim_{x \to - 0.25} 4}{lim_{x \to - 0.25} x + 1}}$

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $- 0.25$ .

$- \sqrt{\frac{lim_{x \to - 0.25} x + 4}{lim_{x \to - 0.25} x + 1}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 0.25$ .

$- \sqrt{\frac{lim_{x \to - 0.25} x + 4}{lim_{x \to - 0.25} x + lim_{x \to - 0.25} 1}}$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $- 0.25$ .

$- \sqrt{\frac{lim_{x \to - 0.25} x + 4}{lim_{x \to - 0.25} x + 1}}$

Step 8

Evaluate the limits by plugging in $- 0.25$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- 0.25$ for $x$ .

$- \sqrt{\frac{- 0.25 + 4}{lim_{x \to - 0.25} x + 1}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 0.25$ for $x$ .

$- \sqrt{\frac{- 0.25 + 4}{- 0.25 + 1}}$

Step 9

Simplify the answer.

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Step 9.1

Add $- 0.25$ and $4$ .

$- \sqrt{\frac{3.75}{- 0.25 + 1}}$

Step 9.2

Add $- 0.25$ and $1$ .

$- \sqrt{\frac{3.75}{0.75}}$

Step 9.3

Divide $3.75$ by $0.75$ .

$- \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{5}$

Decimal Form:

$- 2.23606797 \dots$