Review the table of values for function [tex]g(x)[/tex].
| 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 [tex]\lim _{x \rightarrow 29} g(x)[/tex], if it exists?
A. -4.25
B. -4
C. -3.75
D. DNE
Answer (2)
By analyzing the values of g ( x ) as x approaches 29 from both the left and right, we found that both limits equal -4. Therefore, lim x → 29 g ( x ) = − 4 , making option B the correct answer.
;
Analyze the values of g ( x ) as x approaches 29 from the left.
Analyze the values of g ( x ) as x approaches 29 from the right.
Determine that both limits approach -4.
Conclude that lim x → 29 g ( x ) = − 4 .
Explanation
Analyze the problem We are given a table of values for the function g ( x ) as x approaches 29 from both sides. We want to find the limit of g ( x ) as x approaches 29.
Examine the values To find the limit, we need to examine the values of g ( x ) as x gets closer to 29 from both the left and the right. If the values approach the same number from both sides, then the limit exists and is equal to that number.
Limit from the left As x approaches 29 from the left (i.e., x < 29 ), the values of g ( x ) are -3.751, -3.76, and -3.9. These values appear to be approaching -4. So, we can estimate that lim x → 2 9 − g ( x ) = − 4 .
Limit from the right As x approaches 29 from the right (i.e., 29"> x > 29 ), the values of g ( x ) are -4.1, -4.24, and -4.249. These values also appear to be approaching -4. So, we can estimate that lim x → 2 9 + g ( x ) = − 4 .
Conclusion Since the limit from the left and the limit from the right are both equal to -4, we can conclude that the limit of g ( x ) as x approaches 29 is -4.
Examples
In manufacturing, understanding limits is crucial for quality control. Imagine a machine producing parts, and g ( x ) represents the size of the parts as the machine setting x approaches a target value. If lim x → 29 g ( x ) = − 4 , it means that as the machine setting gets closer to 29, the part size approaches -4 units. This helps engineers fine-tune the machine to produce parts within acceptable tolerances, ensuring consistent quality and preventing defects.
