Graph this a) 

f

(

−

2

)

=

0

f(−2)=0

b) 

lim

⁡

x

→

−

2

−

f

(

x

)

=

4

lim 

x→−2 

−
 

 f(x)=4

c) 

lim

⁡

x

→

−

2

+

f

(

x

)

=

0

lim 

x→−2 

+
 

 f(x)=0

d) 

lim

⁡

x

→

−

2

f

(

x

)

lim 

x→−2

 f(x) does not exist.

e) 

f

(

−

4

)

=

4

f(−4)=4

f) 

lim

⁡

x

→

−

4

−

f

(

x

)

=

4

lim 

x→−4 

−
 

 f(x)=4

g) 

lim

⁡

x

→

−

4

+

f

(

x

)

=

4

lim 

x→−4 

+
 

 f(x)=4

h) 

lim

⁡

x

→

−

4

f

(

x

)

=

4

lim 

x→−4

 f(x)=4

Here is the graph of \( f(x) \) based on the conditions provided:

- \( f(-2) = 0 \).
- \( \lim_{x \to -2^-} f(x) = 4 \), meaning as \( x \) approaches \(-2\) from the left, \( f(x) \) approaches 4.
- \( \lim_{x \to -2^+} f(x) = 0 \), meaning as \( x \) approaches \(-2\) from the right, \( f(x) \) approaches 0.
- \( \lim_{x \to -2} f(x) \) does not exist because the left and right limits are different.
- \( f(-4) = 4 \).
- \( \lim_{x \to -4^-} f(x) = 4 \), meaning as \( x \) approaches \(-4\) from the left, \( f(x) \) remains 4.
- \( \lim_{x \to -4^+} f(x) = 4 \), meaning as \( x \) approaches \(-4\) from the right, \( f(x) \) also remains 4.
- \( \lim_{x \to -4} f(x) = 4 \), so the limit at \(-4\) exists.

Would you like more details on the graph or any of the limit concepts?

Here are 5 related questions to consider:
1. How do left-hand and right-hand limits affect the overall limit at a point?
2. What happens if the left-hand and right-hand limits do not match?
3. Can you find similar examples where a limit does not exist at certain points?
4. How do asymptotes relate to discontinuities in functions?
5. What is the role of piecewise functions in modeling real-world problems?

**Tip:** When dealing with limits, always check both the left-hand and right-hand behavior to determine if the overall limit exists.

Graph this a) f(-2) = 0 b) lim x→−2− f(x) = 4 c) lim x→−2+ f(x) = 0 d) lim x→−2 f(x) does not exist. e) f(−4) = 4 f) lim x→−4− f(x) = 4 g) lim x→−4+ f(x) = 4 h) lim x→−4 f(x) = 4

Explore the graph of a function where limits exist at x = -4 and differ at x = -2. This calculus problem involves understanding left-hand and right-hand limits, discontinuities, and the conditions for the existence of a limit. Suitable for students in Grades 11-12.

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