select the correct answer. what does this function show at (x = 1)? (f(x)=\begin{cases}x + 3&x<1\\3x +…

Answer:

B. jump discontinuity

Explanation:

Step1: Calculate left - hand limit

$\lim_{x\rightarrow1^{-}}f(x)=\lim_{x\rightarrow1^{-}}(x + 3)=1+3 = 4$

Step2: Calculate right - hand limit

$\lim_{x\rightarrow1^{+}}f(x)=\lim_{x\rightarrow1^{+}}(3x + 1)=3\times1+1=4$

Step3: Calculate function value at $x = 1$

$f(1)=3\times1 + 1=4$

Step4: Analyze the one - sided limits and function value

The left - hand limit $\lim_{x\rightarrow1^{-}}f(x)=4$, the right - hand limit $\lim_{x\rightarrow1^{+}}f(x)=4$, but the function changes its rule at $x = 1$. The two parts of the piece - wise function approach different values from different sides of $x = 1$ in a non - smooth way, which is a jump discontinuity.