Limit of Function at x=0: Does Not Exist (DNE)

In summary, the conversation discusses evaluating a limit and determining whether the answer is -∞, ∞, or DNE. The speaker follows the graph and concludes that the answer is DNE, but it is marked as incorrect. The expert explains that technically, the limit does not exist, but it should be marked as ∞ based on the instructions.
  • #1
josesalazmat
2
0
hello
I have an exercise which says:

Evaluate the following limit. Enter -I if your answer is −∞, enter I if your answer is ∞, and enter DNE if the limit does not exist.

\(\displaystyle limx→0[(1/(7x)−(1)/((e^(7x))−1)] \) e power 7x

when I follow the graph for \(\displaystyle 1/7x\) the limit does not exist (goes to infinite for the right and -infinite for the left)it is the same for \(\displaystyle 1/(((e^(7x))−1)\)

my answer is DNE but it is wrong

where is a mistake?

Thanks

Jose
 
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  • #2
Let:

\(\displaystyle f(x)=\frac{\dfrac{1}{7x}-1}{e^{7x}-1}\)

We see that:

\(\displaystyle \lim_{x\to0^{-}}f(x)=\lim_{x\to0^{+}}f(x)=\infty\)

Technically, this limit does not exist, but given the instructions, I would answer with "I."
 

1. What does it mean for the limit of a function at x=0 to not exist?

When the limit of a function at x=0 does not exist, it means that as the value of x approaches 0, the function does not approach a single, finite value. This could be due to a discontinuity or a vertical asymptote in the graph of the function at x=0.

2. How do you determine if the limit of a function at x=0 does not exist?

To determine if the limit of a function at x=0 does not exist, you must evaluate the left-hand and right-hand limits separately. If the left-hand limit is not equal to the right-hand limit, or if either of them is undefined, then the limit at x=0 does not exist.

3. Can a function have a limit at x=0 that does not exist but still be continuous?

Yes, it is possible for a function to be continuous at x=0 and still have a limit that does not exist. This can occur when the function has a removable discontinuity at x=0, meaning that the function can be redefined at that point to make it continuous, but the limit still does not exist.

4. What are some common reasons for a function to have a limit that does not exist at x=0?

Some common reasons for a function to have a limit that does not exist at x=0 include a jump discontinuity, an infinite oscillation, or a vertical asymptote. These can occur when there is a sudden change or a discontinuity in the graph of the function at x=0.

5. Is it possible for a function to have a limit at x=0 that does not exist but still be defined at x=0?

Yes, it is possible for a function to have a limit at x=0 that does not exist and still be defined at x=0. This can occur when the function has a removable discontinuity or a vertical asymptote at x=0, meaning that the function is defined at x=0 but the limit does not exist due to the discontinuity.

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