Find the limit and discuss the continuity of the function. \lim_(x, y)\rightarrow (\frac{\pi}{4}, 2) y\cos xy

Evaluating trig limit ${\displaystyle \underset{x\to 0}{lim}\frac{\sqrt{1-\cos \left(x^2\right)}}{1-\cos \left(x\right)}}$
I have tried to simplify the expression using the identity ${\displaystyle 1-\cos \left(x\right)=2{\sin }^2\left(\frac{x}{2}\right)}$, but I have still failed to remove the indeterminate form.

The limit as x approaches zero of ${\displaystyle \frac{\sin x-x\cos x}{x^3}}$

I want to calculate the limit of
$\underset{x\to 0}{lim}\frac{1-\cosh (x^5)}{x^6{\sin }^2(x^2)}$

Use continuity to evaluate the limit.
$\underset{x\to 3}{lim}\arctan \left(\frac{x^2-9}{5x^2-15x}\right)$
Factored and got...
$\underset{x\to 3}{lim}\arctan \left(\frac{x+3}{5x}\right)$

Find the limit or show that it does not exist.
${\displaystyle \underset{x\to \mathrm{\infty }}{lim}=\frac{1-x^2}{x^3-x+1}}$

Use the limit definition of the derivative to calculate the derivatives of the following function:
(a) $f(x)=4x^2+3x+1$
(b) $f(x)=\frac{2}{x^2}$

How to prove that ${\int }_0^{\mathrm{\infty }}\cos (\alpha t)e^{-\lambda t}\mathrm{d}t=\frac{\lambda }{{\alpha }^2+{\lambda }^2}$ using real methods?