Question

attempt 1 4 attempts remaining evaluate the following limit. give an exact answer if the limit is a number. otherwise, enter -∞ or ∞ if the limit is infinite, or enter dne if the limit does not exist in another way. \\(\lim_{x \to 2} \frac{\sqrt{x + 2} - 2}{x - 2}\\)

Explanation:

Step1: Rationalize the numerator

Multiply numerator and denominator by the conjugate $\sqrt{x+2}+2$:

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

Step2: Simplify the numerator

Use the difference of squares $(a-b)(a+b)=a^2-b^2$:

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

Step3: Cancel common factors

Cancel $(x-2)$ from numerator and denominator (valid since $x
eq 2$ when taking the limit):

$$ \lim_{x \to 2} \frac{1}{\sqrt{x+2}+2} $$

Step4: Substitute $x=2$

Evaluate the limit by direct substitution:

$$ \frac{1}{\sqrt{2+2}+2} = \frac{1}{2+2} $$

Answer:

$\frac{1}{4}$