for the function g whose graph is given, state the value of each quantity, if it exists. (if an answer does…

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.)\n(a) (lim_{t \to 0^{-}} g(t))\n(b) (lim_{t \to 0^{+}} g(t))\n(c) (lim_{t \to 0} g(t))\n(d) (lim_{t \to 2^{-}} g(t))

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.)\n(a) (lim_{t \to 0^{-}} g(t))\n(b) (lim_{t \to 0^{+}} g(t))\n(c) (lim_{t \to 0} g(t))\n(d) (lim_{t \to 2^{-}} g(t))

Answer

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)\neq\lim\limits_{t\to0^{+}}g(t)) (because (3\neq1)), 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)