Mathematics

Problem

Consider the piecewise‐defined function

g(x) = \begin{cases} 3x+1, & \text{if } x < 2, \\ \sqrt{x+5}, & \text{if } x \ge 2 \end{cases}

Answer the following questions about the continuity of

g

. (a) Describe, in precise mathematical language, what it means for a function

h

to be continuous at the point

x=a

. (b) Use the description you provided in part (a) to determine whether

g

is continuous at

x=2

. Include all relevant limit statements and a clear explanation of your reasoning.