For the function g whose graph is shown, find a number a that satisfies the given description.

The x y-coordinate plane is given.
The function enters the window in the third quadrant, goes up and right becoming more steep, crosses the y-axis at approximately y = −0.5, crosses the x-axis at approximately x = 0.5, goes up and right becoming less steep, and stops at the approximate open point (2, 2.75).
The function starts again at the approximate open point (2, 1.75), goes down and right becoming less steep, and stops at the approximate open point (4, 0.5).
The functions starts again at the approximate closed point (4, 2.25), goes down and right becoming more steep, passes through the approximate open point (5, 1.75), crosses the x-axis at approximately x = 6.25, and exits the window in the fourth quadrant.
(a)
lim 
x→a 
g(x)
 does not exist but g(a) is defined.
a = 
(b)
lim 
x→a 
g(x)
 exists but g(a) is not defined.
a = 
(c)
lim 
x→a− 
g(x)
 and 
lim 
x→a+ 
g(x)
 both exist but 
lim 
x→a 
g(x)
 does not exist.
smaller value	a	=	
larger value	a	=	
(d)
lim 
x→a+ 
g(x) = g(a)
 but 
lim 
x→a− 
g(x) ≠ g(a).
a = 



To solve this problem, we need to analyze the graph of the function \( g(x) \) based on the provided description and identify different scenarios for limits and continuity. Let's address each part step by step.

### (a) \(\lim_{x \to a} g(x)\) does not exist, but \( g(a) \) is defined.
This condition describes a point where the function is discontinuous, but the function value is still defined. A common situation is when there is a jump discontinuity or a break in the graph.

From the description:
- At \( x = 4 \), the graph has an open point at \( (4, 0.5) \) and a closed point at \( (4, 2.25) \). This means that \( g(4) \) is defined as 2.25, but the limit does not exist because the left-hand and right-hand limits do not match.

Thus:
\[ a = 4 \]

### (b) \(\lim_{x \to a} g(x)\) exists, but \( g(a) \) is not defined.
This condition indicates that the limit exists, but there is a hole (open point) in the graph at \( x = a \).

From the description:
- At \( x = 2 \), there is a discontinuity where the graph has open points at both \( (2, 2.75) \) and \( (2, 1.75) \). So, the limit exists (since both sides converge to a value), but the function is not defined at \( x = 2 \).

Thus:
\[ a = 2 \]

### (c) \(\lim_{x \to a^-} g(x)\) and \(\lim_{x \to a^+} g(x)\) both exist, but \(\lim_{x \to a} g(x)\) does not exist.
This condition describes a jump discontinuity where the left-hand and right-hand limits both exist but are not equal.

From the description:
- At \( x = 4 \), the left-hand limit approaches 2.25, and the right-hand limit approaches 0.5. Since these limits do not match, the overall limit does not exist.

Thus:
- Smaller value: \( a = 4 \)

At \( x = 2 \), there is also a jump discontinuity, with left and right limits approaching 2.75 and 1.75, respectively. So the limit does not exist at \( x = 2 \) either.

Thus:
- Larger value: \( a = 2 \)

### (d) \(\lim_{x \to a^+} g(x) = g(a)\) but \(\lim_{x \to a^-} g(x) \neq g(a)\).
This condition describes a point where the right-hand limit equals the function value, but the left-hand limit does not.

From the description:
- At \( x = 4 \), the right-hand limit approaches 2.25, which is equal to the value of \( g(4) = 2.25 \), but the left-hand limit is 0.5, so they do not match.

Thus:
\[ a = 4 \]

### Final Answers:
- (a) \( a = 4 \)
- (b) \( a = 2 \)
- (c) Smaller value: \( a = 4 \), Larger value: \( a = 2 \)
- (d) \( a = 4 \)

Let me know if you'd like further details or have any questions!

---

Here are 5 follow-up questions:
1. What is the difference between a jump discontinuity and a removable discontinuity?
2. How do we define a limit at a point when the function has a hole?
3. Why does the existence of a limit at a point not guarantee that the function is continuous there?
4. What are left-hand and right-hand limits, and how do they affect the overall limit at a point?
5. How can we tell if a function is defined at a point from its graph?

**Tip:** A function can have different types of discontinuities: jump, removable, and infinite, each affecting the limits in distinct ways.

For the function g whose graph is shown, find a number a that satisfies the given description.
The x y-coordinate plane is given. The function enters the window in the third quadrant, goes up and right becoming more steep, crosses the y-axis at approximately y = −0.5, crosses the x-axis at approximately x = 0.5, goes up and right becoming less steep, and stops at the approximate open point (2, 2.75). The function starts again at the approximate open point (2, 1.75), goes down and right becoming less steep, and stops at the approximate open point (4, 0.5). The functions starts again at the approximate closed point (4, 2.25), goes down and right becoming more steep, passes through the approximate open point (5, 1.75), crosses the x-axis at approximately x = 6.25, and exits the window in the fourth quadrant.
(a)
lim 
x→a 
g(x)
 does not exist but g(a) is defined.
a =
(b)
lim 
x→a 
g(x)
 exists but g(a) is not defined.
a =
(c)
lim 
x→a− 
g(x)
 and 
lim 
x→a+ 
g(x)
 both exist but 
lim 
x→a 
g(x)
 does not exist.
smaller value
a=
larger value
a=
(d)
lim 
x→a+ 
g(x) = g(a)
 but 
lim 
x→a− 
g(x) ≠ g(a).
a=

Analyze the function g(x) using limits and graph behavior. Learn how to identify points where the limit exists but the function is undefined, and where discontinuities like jump discontinuities and removable discontinuities occur. Suitable for students studying precalculus or calculus.

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