Question

for the function g whose graph is given, state the value of each quantity, if it exists. (if an answer does not exist, enter dne.)
(a) (lim_{t \to 0^{-}} g(t))
(b) (lim_{t \to 0^{+}} g(t))
(c) (lim_{t \to 0} g(t))
(d) (lim_{t \to 2^{-}} g(t))

Explanation:

Step1: Analyze $\lim\limits_{t\to0^{-}}g(t)$

As \(t\) approaches \(0\) from the left (\(t\to0^{-}\)), we look at the part of the graph where \(t < 0\). The \(y -\) value that the graph approaches is \(3\). So, \(\lim\limits_{t\to0^{-}}g(t)=3\).

Step2: Analyze $\lim\limits_{t\to0^{+}}g(t)$

As \(t\) approaches \(0\) from the right (\(t\to0^{+}\)), we look at the part of the graph where \(t>0\). The \(y -\) value that the graph approaches is \(1\). So, \(\lim\limits_{t\to0^{+}}g(t) = 1\).

Step3: Analyze $\lim\limits_{t\to0}g(t)$

Since \(\lim\limits_{t\to0^{-}}g(t)
eq\lim\limits_{t\to0^{+}}g(t)\) (because \(3
eq1\)), by the definition of the limit \(\lim\limits_{t\to a}f(t)\) exists if and only if \(\lim\limits_{t\to a^{-}}f(t)=\lim\limits_{t\to a^{+}}f(t)\), we have \(\lim\limits_{t\to0}g(t)\) does not exist.

Step4: Analyze $\lim\limits_{t\to2^{-}}g(t)$

As \(t\) approaches \(2\) from the left (\(t\to2^{-}\)), we look at the part of the graph where \(t < 2\). The \(y -\) value that the graph approaches is \(2\). So, \(\lim\limits_{t\to2^{-}}g(t)=2\).

Answer:

(a) \(3\)
(b) \(1\)
(c) DNE
(d) \(2\)