Review the table of values for function [tex]h(x)[/tex].
| x | h(x) |
| :---- | :---- |
| 8.9 | 3.83 |
| 8.99 | -1.19 |
| 8.999 | 4.73 |
| 9 | undefined |
| 9.001 | -4.73 |
| 9.01 | 1.19 |
| 9.1 | -3.83 |
Which statement correctly explains whether [tex]\lim _{x \rightarrow 9} h(x)[/tex] exists?
A. The limit does not exist because the values of [tex]h(x)[/tex] seem to oscillate between random values around [tex]x=[/tex] 9.
B. The limit does exist because [tex]h(x)[/tex] is defined for all given values around [tex]x=9[/tex], even though [tex]h(x)[/tex] isn't defined at [tex]x=9[/tex].
C. The limit does not exist because [tex]h(x)[/tex] is not defined at [tex]x=9[/tex], and for a limit to exist, the function must be defined at [tex]x=9[/tex].
D. The limit does exist because the values of [tex]h(x)[/tex] to the right of [tex]x=9[/tex] are all opposites of the values of [tex]h(x)[/tex] to the left of [tex]x=9[/tex].
Answer (2)
The limit lim x → 9 h ( x ) does not exist because the function values oscillate and do not approach a single value as x approaches 9 from both sides. The correct answer is option A.
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Analyze the behavior of h ( x ) as x approaches 9 from the left and right.
Observe that the values of h ( x ) oscillate and do not converge to a single value from either side.
Conclude that the limit does not exist because the left-hand and right-hand limits are not equal.
The limit does not exist because the values of h ( x ) seem to oscillate between random values around x = 9 . The limit does not exist because the values of h ( x ) seem to oscillate between random values around x = 9.
Explanation
Analyzing the Problem We are given a table of values for a function h ( x ) and asked to determine if lim x → 9 h ( x ) exists. We need to analyze the behavior of h ( x ) as x approaches 9 from both the left and the right.
Examining the Behavior of h(x) As x approaches 9 from the left (i.e., x < 9 ), the values of h ( x ) are 3.83, -1.19, and 4.73. These values do not seem to converge to a single number. As x approaches 9 from the right (i.e., 9"> x > 9 ), the values of h ( x ) are -4.73, 1.19, and -3.83. These values also do not seem to converge to a single number.
Determining if the Limit Exists For the limit to exist, the function must approach the same value from both sides. In this case, the values of h ( x ) oscillate and do not approach a single value from either side. Therefore, the limit does not exist.
Selecting the Correct Statement The correct statement is: The limit does not exist because the values of h ( x ) seem to oscillate between random values around x = 9 .
Final Answer Therefore, the limit lim x → 9 h ( x ) does not exist because the values of h ( x ) oscillate as x approaches 9.
Examples
In physics, understanding limits is crucial when analyzing the behavior of physical systems near singularities or discontinuities. For example, when studying the electric field near a point charge, the field strength approaches infinity as you get closer to the charge. Determining whether this limit exists and understanding its behavior helps physicists model and predict the behavior of such systems accurately.
