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Thread: Why does limit tend to positve infinity and 0?

  1. #1
    ORT
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    Why does limit tend to positve infinity and 0?

    Hi everyone,

    I have following task:

    Why does limit tend to positve infinity and 0?-screenshot-8-.png

    When I approach from the right, it's positive infinity (which is correct) but why is it 0 when I approach from the left, why isn't it negative infinity?

    Thank you in advance!

    Kind regards!
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  2. #2
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    Re: Why does limit tend to positve infinity and 0?

    Quote Originally Posted by ORT View Post
    Hi everyone,
    I have following task: What task?
    Click image for larger version. 

Name:	Screenshot (8).png 
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    When I approach from the right, it's positive infinity (which is correct) but why is it 0 when I approach from the left, why isn't it negative infinity?
    Thank you in advance!
    You have posted a simple statement. What is the question?
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  3. #3
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    Re: Why does limit tend to positve infinity and 0?

    Your function is $\displaystyle e^{\frac{1}{x-1}}$ and you are asking about the limit as x goes to 1 from above and below. As x approaches 1 from above, x- 1 is positive so we have e to a larger and larger positive value. As x approaches 1 from below, x- 1 is negative so we have e to a "larger and larger" negative power. Letting y= 1/(x- 1), the limit as x approaches 1 from below is the same as $\displaystyle \lim_{y\to -\infty} e^y$. And that limit is 0.
    Thanks from topsquark and ORT
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    Re: Why does limit tend to positve infinity and 0?

    Quote Originally Posted by HallsofIvy View Post
    Your function is $\displaystyle e^{\frac{1}{x-1}}$ and you are asking about the limit as x goes to 1 from above and below. As x approaches 1 from above, x- 1 is positive so we have e to a larger and larger positive value. As x approaches 1 from below, x- 1 is negative so we have e to a "larger and larger" negative power. Letting y= 1/(x- 1), the limit as x approaches 1 from below is the same as $\displaystyle \lim_{y\to -\infty} e^y$. And that limit is 0.
    I wish the we did not have make a guess as to exact statement of a question.
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    Re: Why does limit tend to positve infinity and 0?

    If you want to see clearly why coming from the Left to 1 tends to zero, do this

    take a number less than 1, but almost equal to 1
    for example 0.9

    now substitute in 1/(x - 1)
    when x = 0.9, 1/(x - 1) = 1/(0.9 - 1) = 1 / -0.1 = -10

    now e^(-10) = 0.000045 (which almost zero)

    now take x = 0.99

    after substitution
    e^(-100) = 0.000000000000000000000000000000000000000000037 (wOW it is so clear as x goes to 1 from the left the function becomes zero)

    try x = 0.999 if you want
    Thanks from topsquark
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