Questions: Let f(x) = sqrt(x+1) + 2. Determine the following function value. Enter DNE if the function is undefined at x=-1. f(-1) = 2 Determine the following limit. Enter DNE if the limit fails to exist, except in cart 2 of 4 infinite limit exists, enter ∞ or -∞, as appropriate. lim x -> -1 sqrt(x+1) + 2 = 2 Use the above information to determine whether or not f is continuous at x=-1. f is not continuous at x=-1. f is continuous at x=-1.

Let \(f(x)=\sqrt{|x+1|}+2\). Determine \(f(-1)\).

Evaluate \(f(-1)\)

\(f(-1) = \sqrt{|-1+1|} + 2 = \sqrt{|0|} + 2 = \sqrt{0} + 2 = 0 + 2 = 2\)

\(\boxed{2}\)

Determine \(\lim_{x \rightarrow -1} \sqrt{|x+1|}+2\).

Evaluate the limit.

As \(x\) approaches \(-1\), \(|x+1|\) approaches \(|0|\), which is \(0\). Thus, \(\sqrt{|x+1|}\) approaches \(\sqrt{0} = 0\). Therefore, the limit is \(\lim_{x \rightarrow -1} \sqrt{|x+1|}+2 = 0 + 2 = 2\).

\(\boxed{2}\)

Use the above information to determine whether or not \(f\) is continuous at \(x=-1\).

Determine continuity.

A function \(f(x)\) is continuous at \(x=c\) if \(\lim_{x \rightarrow c} f(x) = f(c)\). We have \(f(-1) = 2\) and \(\lim_{x \rightarrow -1} f(x) = 2\). Since \(\lim_{x \rightarrow -1} f(x) = f(-1)\), the function \(f(x)\) is continuous at \(x=-1\).

\(\boxed{f \text{ is continuous at } x=-1}\)

\(f(-1) = 2\) \(\lim_{x \rightarrow -1} \sqrt{|x+1|}+2=2\) \(f \text{ is continuous at } x=-1\)