Questions: Given that the limit of f(x) as x approaches 3 is -5 and the limit of g(x) as x approaches 3 is 6, find the following limit. The limit of the square root of g(x) - f(x) as x approaches 3 is The limit of the square root of g(x) - f(x) as x approaches 3 is (Simplify your answer. Type an exact answer, using radicals as needed.)

Given that the limit of f(x) as x approaches 3 is -5 and the limit of g(x) as x approaches 3 is 6, find the following limit.
The limit of the square root of g(x) - f(x) as x approaches 3 is
The limit of the square root of g(x) - f(x) as x approaches 3 is

(Simplify your answer. Type an exact answer, using radicals as needed.)

Transcript text: Given that $\lim _{x \rightarrow 3} f(x)=-5$ and $\lim _{x \rightarrow 3} g(x)=6$, find the following limit. \[ \lim _{x \rightarrow 3} \sqrt{g(x)-f(x)} \] \[ \lim _{x \rightarrow 3} \sqrt{g(x)-f(x)}= \] $\square$ (Simplify your answer. Type an exact answer, using radicals as needed.)

Solution

Solution Steps

Step 1: Substitute the given limits into the expression

Given: \[ \lim _{x \rightarrow 3} f(x) = -5 \quad \text{and} \quad \lim _{x \rightarrow 3} g(x) = 6 \] Substitute these values into the expression $ \sqrt{g(x) - f(x)} $: \[ \sqrt{g(x) - f(x)} \rightarrow \sqrt{6 - (-5)} = \sqrt{6 + 5} \]

Step 2: Simplify the expression inside the square root

Simplify the expression inside the square root: \[ \sqrt{6 + 5} = \sqrt{11} \]

Step 3: Write the final limit

The limit is: \[ \lim _{x \rightarrow 3} \sqrt{g(x) - f(x)} = \sqrt{11} \]