Q. 38 - Calculus

Question

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\underset{x\to -5}{\lim } \sqrt{x} .$

Step-by-Step Solution

Answer

Limit does not exist.

Step 1. Given Information.

$$\begin{gathered} Given: \\ \underset{x\to -5}{\lim } \sqrt{x} \end{gathered}$$

Step 2. Theorem 1.16

$$\begin{gathered} Power functions are continuous on their domains. \\ In terms of limits, if A is real and k is rational, \\ then for all values x = c at which x^k is defined we have \\ \underset{x\to c}{\lim } A{x}^k = A{c}^k. \end{gathered}$$

Step 3. Finding limit of given function.

According to the theorem power functions are continuous in their domains but $x=-5$ is not in the domain of the function $\sqrt{x}$ . Therefore, theorem does not apply for this limit and also limit does not exist for this function as it is not even in the domain.