nonzero division by zero for limits at negative infinity

Question

I solved my limit as shown below and I concluded that it does not exist for it is a nonzero divided by zero. But, the given solution states that it does in fact exist and that the limit is positive infinity. I understand that if you have a constant in the numerator and a variable in the denominator and when that variable approaches infinity (be it negative or positive infinity) it will evaluate to zero; BUT in my limit the denominator is a limit of a constant (i.e. the value of this limit is the constant itself) subtracted from a limit of a square root that approaches 1, i.e. 1-1=0 so the whole denominator is zero not because the limit of the whole denominator approaches zero but because of the subtraction between these two terms. I hope this makes as much sense to you as it does to me. If I am wrong please explain to me why, and in case you forgot my answer is that the limit does not exist but the solution states it does exist and that it is positive infinity.

Answer 1

Your mistake is taking the limit of 1 and $\sqrt{1 + 1/x}$ and the fraction as a whole seperately instead of at the same time. $\sqrt{1 + 1/x}$ is slightly smaller than 1 for small (I mean far to the left side of the number line, not close to 0) values of $x$, so $y := 1 - \sqrt{1 + 1/x}$ will tend to $0$ from the right side as $x$ approaches infinity. This rewrites your problem to $\lim_{y \to 0+}\frac{1}{y} = \infty$.