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Thread: Limit exist or not?

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    Limit exist or not?

    Find the following limit, if it exist:
    $\displaystyle \lim_{x \to 1}\frac{x + 2}{x^2 + x - 2}$

    apply L'Hospital's Rule,
    i get, $\displaystyle \frac{1}{3}$

    so the limit is exist..
    correct?
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  2. #2
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    Quote Originally Posted by nameck View Post
    Find the following limit, if it exist:
    $\displaystyle \lim_{x \to 1}\frac{x + 2}{x^2 + x - 2}$

    apply L'Hospital's Rule,
    i get, $\displaystyle \frac{1}{3}$

    so the limit is exist..
    correct?
    You can't apply L'Hospital's rule, because it does not tend to $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$.


    $\displaystyle \frac{x + 2}{x^2 + x - 2} = \frac{x + 2}{(x + 2)(x - 1)}$

    $\displaystyle = \frac{1}{x - 1}$.


    So $\displaystyle \lim_{x \to 1}\frac{x + 2}{x^2 + x - 2} = \lim_{x \to 1}\frac{1}{x - 1}$


    Try graphing this function. I think you'll see that the left hand limit is $\displaystyle -\infty$ and the right hand limit is $\displaystyle \infty$.

    Since the left and right hand limits don't match, the limit does not exist.
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