Question

Please help! thanks so much​

Answer 1

Answer:

See below.

Step-by-step explanation:

First, we can see that $\lim_{x \to 2} (f(x))= -1$.

Thus, for the question, we can just plug -1 in:

$\lim_{x \to 2} (\frac{x}{f(x)+1})=\frac{(2)}{-1+1} =und.$

Saying undefined (or unbounded) will be correct.

However, note that as x approaches 2, the values of y decrease in order to get to -1. In other words, $f(x)$ will always be greater or equal to -1 (you can also see this from the graph). This means that as x approaches 2, f(x) will approach -.99 then -.999 then -.9999 until it reaches -1 and then go back up. What is important is that because of this, we can determine that:

$\lim_{x \to 2} (\frac{x}{f(x)+1})=\frac{(2)}{-1+1} = +\infty$

This is because for the denominator, the +1 will always be greater than the f(x). This makes this increase towards positive infinity. Note that limits want the values of the function as it approaches it, not at it.