Review the table of values for function g(x).
x
g(x)
28.9
–3.751
28.99
–3.76
28.999
–3.9
29
undefined
29.001
–4.1
29.01
–4.24
29.1
–4.249
What is Limit of g (x) as x approaches 29g(x), if it exists?
–4.25
–4
–3.75
DNE
The Correct answer and Explanation is:
The correct answer is –4.
Explanation:
To find the limit of g(x)g(x)g(x) as xxx approaches 29, we need to evaluate the behavior of the function from both sides of x=29x = 29x=29. Limits help us understand the value that a function approaches as the input xxx gets closer and closer to a particular point, even if the function is not defined at that point.
In this case, we are given the following values for g(x)g(x)g(x):
- As xxx approaches 29 from the left:
- At x=28.9x = 28.9x=28.9, g(x)=−3.751g(x) = -3.751g(x)=−3.751.
- At x=28.99x = 28.99x=28.99, g(x)=−3.76g(x) = -3.76g(x)=−3.76.
- At x=28.999x = 28.999x=28.999, g(x)=−3.9g(x) = -3.9g(x)=−3.9.
- As xxx approaches 29 from the right:
- At x=29.001x = 29.001x=29.001, g(x)=−4.1g(x) = -4.1g(x)=−4.1.
- At x=29.01x = 29.01x=29.01, g(x)=−4.24g(x) = -4.24g(x)=−4.24.
- At x=29.1x = 29.1x=29.1, g(x)=−4.249g(x) = -4.249g(x)=−4.249.
We also know that g(x)g(x)g(x) is undefined at exactly x=29x = 29x=29, but that doesn’t prevent the limit from existing. What we are interested in is whether the values of g(x)g(x)g(x) approach a single number as xxx gets closer to 29 from both sides.
From the left-hand side (x→29−x \to 29^-x→29−), the values of g(x)g(x)g(x) are getting closer to approximately −4-4−4. From the right-hand side (x→29+x \to 29^+x→29+), the values of g(x)g(x)g(x) are also getting closer to −4-4−4.
Since the function approaches −4-4−4 from both directions, we conclude that the limit of g(x)g(x)g(x) as xxx approaches 29 is:limx→29g(x)=−4\lim_{{x \to 29}} g(x) = -4x→29limg(x)=−4
The limit exists and is equal to −4-4−4, even though the function is undefined at x=29x = 29x=29. This demonstrates how limits can describe the behavior of a function near a point, even when the function has no value at that point.