Calculus Examples

$lim_{x \to - 0.001} \frac{\sqrt{x + 2} - \sqrt{2}}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 0.001$ .

$\frac{lim_{x \to - 0.001} \sqrt{x + 2} - \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 0.001$ .

$\frac{lim_{x \to - 0.001} \sqrt{x + 2} - lim_{x \to - 0.001} \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 0.001} x + 2} - lim_{x \to - 0.001} \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 0.001$ .

$\frac{\sqrt{lim_{x \to - 0.001} x + lim_{x \to - 0.001} 2} - lim_{x \to - 0.001} \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 5

Evaluate the limit of $2$ which is constant as $x$ approaches $- 0.001$ .

$\frac{\sqrt{lim_{x \to - 0.001} x + 2} - lim_{x \to - 0.001} \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 6

Evaluate the limit of $\sqrt{2}$ which is constant as $x$ approaches $- 0.001$ .

$\frac{\sqrt{lim_{x \to - 0.001} x + 2} - \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 7

Evaluate the limits by plugging in $- 0.001$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 0.001$ for $x$ .

$\frac{\sqrt{- 0.001 + 2} - \sqrt{2}}{lim_{x \to - 0.001} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 0.001$ for $x$ .

$\frac{\sqrt{- 0.001 + 2} - \sqrt{2}}{- 0.001}$

Step 8

Simplify the answer.

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Step 8.1

Add $- 0.001$ and $2$ .

$\frac{\sqrt{1.999} - \sqrt{2}}{- 0.001}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{\sqrt{1.999} - \sqrt{2}}{0.001}$

Step 8.3

Evaluate the root.

$- \frac{1.41385996 - \sqrt{2}}{0.001}$

Step 8.4

Evaluate the root.

$- \frac{1.41385996 - 1 \cdot 1.41421356}{0.001}$

Step 8.5

Multiply $- 1$ by $1.41421356$ .

$- \frac{1.41385996 - 1.41421356}{0.001}$

Step 8.6

Subtract $1.41421356$ from $1.41385996$ .

$- \frac{- 0.00035359}{0.001}$

Step 8.7

Divide $- 0.00035359$ by $0.001$ .

$- - 0.35359759$

Step 8.8

Multiply $- 1$ by $- 0.35359759$ .

$0.35359759$