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 $\lim _{x \rightarrow a^{-}} g(x) \neq g(a)$. $a=$ $\square$
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Solution

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Solution Steps

Step 1: Identify where the limit does not exist but the function is defined
  • We need to find a point a a where limxag(x) \lim_{x \to a} g(x) does not exist, but g(a) g(a) is defined.
  • From the graph, at x=2 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) g(2) is defined.

Final Answer

a=2 a = 2

Step 2: Identify where the limit exists but the function is not defined
  • We need to find a point a a where limxag(x) \lim_{x \to a} g(x) exists, but g(a) g(a) is not defined.
  • From the graph, at x=4 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) g(4) is not defined.
Final Answer

a=4 a = 4

Step 3: Identify where both one-sided limits exist but the limit does not exist
  • We need to find a point a a where limxag(x) \lim_{x \to a^-} g(x) and limxa+g(x) \lim_{x \to a^+} g(x) both exist, but limxag(x) \lim_{x \to a} g(x) does not exist.
  • From the graph, at x=2 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 a = 2 Larger value a=2 a = 2

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