$\lim _{n \rightarrow \infty} P\left(1+\frac{r}{100 n}\right)^{t n} =$

$\mathop {\lim }\limits_{x \to \infty } \,{\left( {\frac{{x + a}}{{x + b}}} \right)^{x + b}} = $

$\lim_{x \rightarrow \frac{\pi}{2}} (\tan^{2} x (\sqrt{2 \sin^{2} x + 3 \sin x + 4} - \sqrt{\sin^{2} x + 6 \sin x + 2}))$ is equal to

Let $[x]$ denote the greatest integer less than or equal to $x$. Then,evaluate the limit: $\mathop {\lim }\limits_{x \to 0} \,\frac{{\tan \,(\pi \,{{\sin }^2}\,x) + \,{{(\left| x \right|\, - \,\sin \,(x\,[x]))}^2}}}{{{x^2}}}$

If a function $f$ is defined by $f(x) = \frac{\cot^3 x - \tan x}{\cos(x + \pi/4)}$ for $x \neq \pi/4$,then $\lim_{x \rightarrow \pi/4} f(x) = $

$\mathop {\lim }\limits_{n \to \infty } \frac{{[{1^2}x + {1^2}] + [{2^2}x + {2^2}] + [{3^2}x + {3^2}] + \dots + [{n^2}x + {n^2}]}}{{{n^3}}}$ is equal to :- (where $[.]$ denotes the greatest integer function)