Question

functions f and g are graphed.
find \\(\lim\limits_{x\to 2} \big(f(x)g(x)\big)\\).
choose 1 answer:
a 0
b 2
c 4
d 8
e the limit doesnt exist.

Explanation:

Step1: Find $\lim_{x\to 2} f(x)$

From the graph of $f(x)$, as $x$ approaches 2 from both left and right, the value of $f(x)$ approaches 0. So $\lim_{x\to 2} f(x) = 0$.

Step2: Find $\lim_{x\to 2} g(x)$

From the graph of $g(x)$, as $x$ approaches 2 from both left and right, the value of $g(x)$ approaches 2 (the open circle at $x = 2$ for $g(x)$ has a $y$-value of 2, and the function approaches this value from both sides). So $\lim_{x\to 2} g(x) = 2$.

Step3: Use the product rule for limits

The product rule for limits states that $\lim_{x\to a} (f(x)g(x))=\lim_{x\to a} f(x)\cdot\lim_{x\to a} g(x)$ (when both limits exist). Substituting the values we found: $\lim_{x\to 2} (f(x)g(x))=\lim_{x\to 2} f(x)\cdot\lim_{x\to 2} g(x)=0\times2 = 0$.

Answer:

A. 0