Let $g(x)=\begin{cases} \dfrac{\sin x}{x}, & x<5,\\ m, & x=5,\\ \dfrac{x-5}{\sqrt{x-5}}, & x>5, \end{cases}$ …

Mathematics

Problem

Let

g(x)=\begin{cases} \dfrac{\sin x}{x}, & x<5,\\ m, & x=5,\\ \dfrac{x-5}{\sqrt{x-5}}, & x>5, \end{cases}

where

m

is a real constant. Answer the following questions about the continuity of

g

. a. Part 1: (a) Compute

\displaystyle \lim_{x\to5^-} g(x)

. Show all steps in your evaluation. b. Part 2: (b) Compute

\displaystyle \lim_{x\to5^+} g(x)

. Show all steps, including any algebraic simplifications or limit properties used. c. Part 3: (c) Determine the value of

m

that makes

g

continuous at

x=5

. Justify your result by explicitly connecting the three conditions of continuity at a point.