Calculate the limit: $\lim_{x \to1} \frac{1}{|x^2-1|}$

Question

Calculate the limit $$ \lim_{x \to1} \frac{1}{|x^2-1|} $$ Could you please tell me how to know if 1/0 is positive or minus infinity?

Answer 1

For $ x \ne 1$, $\frac{1}{|x^2-1|}>0$, hence $\lim_{x \to1} \frac{1}{|x^2-1|}= + \infty$

Answer 2

If you know the form is $1/0$ then all you have to ask yourself is: Is the quantity in the faction positive or negative; in this specific case, since there is an absolute value, it is clearly never negative. However, the general approach would look like the following:

Case 1: $x>1$ (but very close to 1). In this case $x^2 > 1$ so $x^2-1>0$ and the fraction is positive. This shows you that

$$\lim_{x\to 1^{+}}\frac{1}{|x^2 - 1|} = +\infty$$

Case 2: $x<1$ (but very close to 1). In this case $x^2 < 1$ so $x^2-1<0$. However, in your case there is an absolute value, and so the denominator is positive. This shows that

$$\lim_{x\to 1^{-}}\frac{1}{|x^2-1|} = +\infty$$

And you are done. Note that in general, this is the approach you want to look at (look at both one-sided limits).