Questions: For the function g whose graph is shown, find a number a that satisfies the given description. (a) The limit of g(x) as x approaches a does not exist but g(a) is defined. a=2 (b) The limit of g(x) as x approaches a exists but g(a) is not defined. a= (c) The limit of g(x) as x approaches a from the left and the limit of g(x) as x approaches a from the right both exist but the limit of g(x) as x approaches a does not exist. smaller value a= larger value a= (d) The limit of g(x) as x approaches a from the right equals g(a) but the limit of g(x) as x approaches a from the left does not equal g(a). a=

For the function g whose graph is shown, find a number a that satisfies the given description.
(a) The limit of g(x) as x approaches a does not exist but g(a) is defined.
a=2
(b) The limit of g(x) as x approaches a exists but g(a) is not defined.
a=
(c) The limit of g(x) as x approaches a from the left and the limit of g(x) as x approaches a from the right both exist but the limit of g(x) as x approaches a does not exist.
smaller value a=
larger value a=
(d) The limit of g(x) as x approaches a from the right equals g(a) but the limit of g(x) as x approaches a from the left does not equal g(a).
a=

Transcript text: For the function $g$ whose graph is shown, find a number a that satisfies the given description. (a) $\lim _{x \rightarrow a} g(x)$ does not exist but $g(a)$ is defined. \[ a=2 \] (b) $\lim _{x \rightarrow a} g(x)$ exists but $g(a)$ is not defined. $a=$ $\qquad$ (c) $\lim _{x \rightarrow a^{-}} g(x)$ and $\lim _{x \rightarrow a^{+}} g(x)$ both exist but $\lim _{x \rightarrow a} g(x)$ does not exist. smaller value $a=$ $\qquad$ larger value $\quad a=$ $\qquad$ (d) $\lim _{x \rightarrow a^{+}} g(x)=g(a)$ but $\lim _{x \rightarrow a^{-}} g(x) \neq g(a)$. $a=$ $\square$

Solution

Solution Steps

Step 1: Identify where the limit does not exist but the function is defined

We need to find a point $ a $ where $ \lim_{x \to a} g(x) $ does not exist, but $ g(a) $ is defined.
From the graph, at $ x = 2 $, the function has a jump discontinuity, meaning the left-hand limit and right-hand limit are not equal, so the limit does not exist. However, $ g(2) $ is defined.

Final Answer

$ a = 2 $

Step 2: Identify where the limit exists but the function is not defined

We need to find a point $ a $ where $ \lim_{x \to a} g(x) $ exists, but $ g(a) $ is not defined.
From the graph, at $ x = 4 $, the function has a hole, meaning the limit exists as the left-hand limit and right-hand limit are equal, but $ g(4) $ is not defined.

Final Answer

$ a = 4 $

Step 3: Identify where both one-sided limits exist but the limit does not exist

We need to find a point $ a $ where $ \lim_{x \to a^-} g(x) $ and $ \lim_{x \to a^+} g(x) $ both exist, but $ \lim_{x \to a} g(x) $ does not exist.
From the graph, at $ x = 2 $, the left-hand limit and right-hand limit exist but are not equal, so the overall limit does not exist.

Final Answer

Smaller value $ a = 2 $ Larger value $ a = 2 $

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