Evaluate lim e^x when x approaches zero from negative

In summary, when x approaches zero from the negative side, the function e^x is not continuous, but if x is approached from the positive side then the function is continuous.
  • #1
Outrageous
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Homework Statement


What is the limit of e^x when x approaches zero from negative side


Homework Equations





The Attempt at a Solution


Taylor series? Then the answer is put all x= 0 , and the answer is 1, but why the question ask from negative side??

Thank you very much
 

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  • #2
Can you use that e^x is continuous? This would allow to use e^0.
If not, what do you know about the exponential function?

That is a strange question.
 
  • #3
The question didn't say, but if it is continuous then I can use Taylor series for e^x . Then just substitute all x with zero? And get answer 1 . Then what is the point to have x approaches zero? Or should I use graph ?
Thank you
 
  • #4
Outrageous said:
The question didn't say, but if it is continuous then I can use Taylor series for e^x . Then just substitute all x with zero? And get answer 1 . Then what is the point to have x approaches zero? Or should I use graph ?
Thank you

What are the *definitions* of e and e^x that you are allowed to use? What facts about e^x are you allowed to use? The point is: how you deal with the problem depends crucially on what properties of e^x you know already. The question you present is almost meaningless, because you leave out so much important information.
 
  • #5
Outrageous said:
The question didn't say, but if it is continuous then I can use Taylor series for e^x .
It is not sufficient for a function to be continuous in order that it HAVE a Taylor series. But the Taylor series has nothing to do with the question.

Then just substitute all x with zero? And get answer 1 . Then what is the point to have x approaches zero? Or should I use graph ?
Thank you
Do you understand what "continuous" means? The definition of "continuous" is that that the limit, as x goes to a, is equal to the value of the function, f(a). That is the point of "x approaches 0"- you evaluate the function at x= 0. If you meant to ask "why approach 0 from the negative side" there doesn't appear to be any special reason except perhaps to see if you really understood the idea of "limit". If f(x) is continuous at x= a (and e^x is continuous for all x), then, by definition, [itex]\lim_{x\to a} f(x)= f(a)[/itex], and, if the limit exists, [itex]\lim_{x\to a^-}f(x)= \lim_{x\to a^+} f(x)= \lim_{x\to a} f(x)[/itex].
 
  • #6
Well e^x is differentiable therefore continuous at 0. So the limit at left of 0 is the same as the right of 0.
 
  • #7
Thank you guys.

This is the whole question that I get from past year exam .
Evaluate lim e^x when x approaches zero from negative.

HallsofIvy said:
Do you understand what "continuous" means? The definition of "continuous" is that that the limit, as x goes to a, is equal to the value of the function, f(a). That is the point of "x approaches 0"- you evaluate the function at x= 0. If you meant to ask "why approach 0 from the negative side" there doesn't appear to be any special reason except perhaps to see if you really understood the idea of "limit". If f(x) is continuous at x= a (and e^x is continuous for all x), then, by definition, [itex]\lim_{x\to a} f(x)= f(a)[/itex], and, if the limit exists, [itex]\lim_{x\to a^-}f(x)= \lim_{x\to a^+} f(x)= \lim_{x\to a} f(x)[/itex].

Thank you . I am too eager to solve problem until forget all the important basic knowledge.
 

Related to Evaluate lim e^x when x approaches zero from negative

1. What is the value of the limit of e^x as x approaches zero from the negative side?

The value of the limit of e^x as x approaches zero from the negative side is 1.

2. How do you evaluate the limit of e^x as x approaches zero from the negative side?

To evaluate the limit of e^x as x approaches zero from the negative side, you can use the rule that states that e^x is continuous everywhere and therefore the limit of e^x as x approaches a is equal to e^a.

3. Why is the value of the limit of e^x as x approaches zero from the negative side equal to 1?

The value of the limit of e^x as x approaches zero from the negative side is equal to 1 because e^x is an exponential function with a base of e, which is a constant value. Therefore, as x approaches zero from the negative side, the value of e^x will always approach 1.

4. Can you use the L'Hopital's rule to evaluate the limit of e^x as x approaches zero from the negative side?

Yes, you can use the L'Hopital's rule to evaluate the limit of e^x as x approaches zero from the negative side. The rule states that if the limit of f(x)/g(x) as x approaches a is in an indeterminate form (such as 0/0 or ∞/∞), then the limit is equal to the limit of the derivative of f(x) divided by the derivative of g(x).

5. How does the value of the limit of e^x as x approaches zero from the negative side differ from the value of the limit as x approaches zero from the positive side?

The value of the limit of e^x as x approaches zero from the negative side is the same as the value of the limit as x approaches zero from the positive side. This is because e^x is an even function, meaning that its graph is symmetrical about the y-axis. Therefore, as x approaches zero from both the negative and positive sides, the value of e^x will always approach 1.

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