Question

graph the following function and then find the specified limit. when necessary, state that the limit does not exist.
g(x)=\begin{cases}x, & \text{for }x < 0\\x^{2}, & \text{for }x>0end{cases} find (lim_{x
ightarrow0}g(x))
choose the correct graph below

Explanation:

Step1: Find left - hand limit

For \(x<0\), \(G(x)=x\). So, \(\lim_{x
ightarrow0^{-}}G(x)=\lim_{x
ightarrow0^{-}}x = 0\)

Step2: Find right - hand limit

For \(x > 0\), \(G(x)=x^{2}\). So, \(\lim_{x
ightarrow0^{+}}G(x)=\lim_{x
ightarrow0^{+}}x^{2}=0\)

Step3: Determine the limit

Since \(\lim_{x
ightarrow0^{-}}G(x)=\lim_{x
ightarrow0^{+}}G(x) = 0\), then \(\lim_{x
ightarrow0}G(x)=0\)

The graph of \(y = x\) for \(x<0\) is a straight - line with slope \(1\) and \(y = x^{2}\) for \(x>0\) is a parabola opening upwards. When \(x = 0\), the function is not defined at \(x = 0\) itself, but the left - hand and right - hand limits are equal.

The correct graph is the one where the line \(y = x\) approaches \(0\) from the left of \(x = 0\) and the parabola \(y=x^{2}\) approaches \(0\) from the right of \(x = 0\) with an open - circle at \(x = 0\).

Answer:

The limit \(\lim_{x
ightarrow0}G(x)=0\)