Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{{(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 4 x + lim_{x \to 2} 4}}{x - 2}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + lim_{x \to 2} 4}}{x - 2}$

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{\sqrt{{(lim_{x \to 2} x)}^{2} - 4 lim_{x \to 2} x + 4}}{x - 2}$

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

$\frac{\sqrt{2^{2} - 4 \cdot 2 + 4}}{x - 2}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Raise $2$ to the power of $2$ .

$\frac{\sqrt{4 - 4 \cdot 2 + 4}}{x - 2}$

Step 7.1.2

Multiply $- 4$ by $2$ .

$\frac{\sqrt{4 - 8 + 4}}{x - 2}$

Step 7.1.3

Subtract $8$ from $4$ .

$\frac{\sqrt{- 4 + 4}}{x - 2}$

Step 7.1.4

Add $- 4$ and $4$ .

$\frac{\sqrt{0}}{x - 2}$

Step 7.1.5

Rewrite $0$ as $0^{2}$ .

$\frac{\sqrt{0^{2}}}{x - 2}$

Step 7.1.6

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

$\frac{0}{x - 2}$

Step 7.2

Divide $0$ by $x - 2$ .

$0$