Q:

Limit of a trigonometric rational expression

How to evaluate the limit of this expression?
$$\lim_{x\to\infty} \frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$$
I managed to simplify the denominator into a sinus form by the Pythagorean formula, and also modified the argument of the sinus in the numerator by dividing with a conjugate which I think is necessary, into this final form:
$$\lim_{x\to\infty} \frac{\sin^2\left(\frac{1}{\sqrt{x+1}+\sqrt{x}}\right)}{\sin^2\frac{1}{x}}$$
And that's where I got stuck. So far I've tried to come up with some solution to eliminate the sinus from the denominator, without success. I'm sure the solution is pretty obvious but I don't quite know how to modify the trigonometric functions in this fashion, so any help would be most appreciated.

A:

Observe
\begin{align}
\lim_{x\to\infty}\frac{\sin^2\left(\sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}&=\lim_{x\to\infty}\frac{\sin^2\left(\sqrt{x+1}-\sqrt{x}\right)}{\sin^2\frac{1}{x}}\\
&=\lim_{x\to\infty}\frac{\left(\sqrt{x+1}-\sqrt{x}\right)^2\frac{\sin^2\left(\sqrt{x+1}-\sqrt{x}\right)}{(\sqrt{x+1}-\sqrt{x})^2}}{\frac{1}{x^2}\frac{\sin^2(1/x)}{1/x^2}}
\end{align}
Now, setting $t=\dfrac{1}{\sqrt{x+1}+\sqrt{x}}$ we note $x\to\infty$ implies $t\to 0^+$, and since $\displaystyle{\lim_{t\to 0}\frac{\sin t}{t}=1}$
$$\lim_{x\to\infty}\frac{\sin^2(\sqrt{x+1}-\sqrt{x})}{(\sqrt{x+1}-\sqrt{x})^2}=\left[\lim_{x\to \infty}\frac{\sin \frac{1}{\sqrt{x+1}+\sqrt{x}}}{\frac{1}{\sqrt{x+1}+\sqrt{x}}}\right]^2=\left(\lim_{t\to 0^+}\frac{\sin t}{t}\right)^2=(1)^2=1$$
Similarly, putting $t=1/x$ we have $t\to0^+$ as $x\to\infty$, and
$$\lim_{x\to\infty}\frac{\sin^2(1/x)}{1/x^2}=\left(\lim_{t\to 0^+}\frac{\sin t}{t}\right)^2=1^2=1$$
Then,
\begin{align}
\lim_{x\to\infty}\frac{\sin^2\left(\sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}&=\left(\lim_{x\to \infty}\frac{\left(\sqrt{x+1}-\sqrt{x}\right)^2}{\frac{1}{x^2}}\right)\lim_{x\to\infty}\frac{\frac{\sin^2\left(\sqrt{x+1}-\sqrt{x}\right)}{(\sqrt{x+1}-\sqrt{x})^2}}{\frac{\sin^2(1/x)}{1/x^2}}\\[5pt]
&=\left(\lim_{x\to\infty}x^2\left(\frac{1}{\sqrt{x+1}+\sqrt{x}}\right)^2\right)\cdot\frac{1}{1}\\
&=\lim_{x\to\infty}\left(\frac{x}{\sqrt{x+1}+\sqrt{x}}\right)^2
\end{align}
You can determine the last limit. 

