Question

evaluate the limit, if it exists. (if an answer does not exist, enter dne.) \\(\lim_{x \to 2} \frac{x^2 + 6x + 4}{x - 2}\\)

Explanation:

Step1: Check direct substitution

Substitute $x=2$ into the function:
$\frac{2^2 + 6\times2 + 4}{2 - 2} = \frac{4 + 12 + 4}{0} = \frac{20}{0}$

Step2: Analyze one-sided limits

Left-hand limit ($x\to2^-$):

As $x$ approaches 2 from the left, $x-2$ approaches 0 from the negative side, so $\frac{x^2 + 6x + 4}{x-2}\to-\infty$.

Right-hand limit ($x\to2^+$):

As $x$ approaches 2 from the right, $x-2$ approaches 0 from the positive side, so $\frac{x^2 + 6x + 4}{x-2}\to+\infty$.
Since the left-hand and right-hand limits are not equal, the two-sided limit does not exist.

Answer:

DNE