1. Evaluate the limit of f(x) as x approaches 1 from the left. 2. Identify the type of discontinuity (removable, jump essential, or infinite essential) of g(x) = 1/f(x) at x = -2.... 1. Evaluate the limit of f(x) as x approaches 1 from the left. 2. Identify the type of discontinuity (removable, jump essential, or infinite essential) of g(x) = 1/f(x) at x = -2. 3. True or False: f satisfies the conditions of the Intermediate Value Theorem on the interval [0, 1]. 4. Fill in the blank: The line with equation _____ is an asymptote of f because the limit of f(x) as x approaches positive infinity is 1. Read more

Steps to Solve

1. Evaluate the limit as $x$ approaches 1 from the left

We need to find the value that $f(x)$ approaches as $x$ gets closer to 1 from values less than 1. Looking at the graph, as $x$ approaches 1 from the left, $f(x)$ approaches -1.

2. Identify the discontinuity of $g(x) = \frac{1}{f(x)}$ at $x = -2$

At $x = -2$, the value of $f(x)$ is zero. This means $g(x) = \frac{1}{f(x)} = \frac{1}{0}$, which tends to infinity. Therefore, $g(x)$ has an infinite essential discontinuity at $x = -2$.

3. Check if $f$ satisfies the Intermediate Value Theorem on $[0, 1]$

The Intermediate Value Theorem (IVT) requires the function to be continuous on the closed interval $[0, 1]$. From the graph, we can see that $f(x)$ is discontinuous at $x = 1$ as it has a "hole". Therefore, $f$ does not satisfy the conditions of the IVT on $[0, 1]$. The statement is FALSE

4. Determine the asymptote of $f$ as $x$ approaches infinity

Given that $\lim_{x \to +\infty} f(x) = 1$, this means that as $x$ goes to positive infinity, the function $f(x)$ approaches 1. This indicates a horizontal asymptote at $y = 1$.