Tamil Nadu Board Solutions for Chapter: Applications of Differential Calculus, Exercise 5: Exercise 5

Evaluate $\underset{x\to \infty }{\lim }\frac{2x^2-3}{x^2-5x+3}$, if necessary use L’Hospital Rule.

Evaluate the following limit, if necessary use L’Hospital Rule :

$\underset{x\to \infty }{\lim }{\mathrm{e}}^{-x}\sqrt{x}$

Evaluate the following limit, if necessary use L’Hospital Rule :

$\underset{x\to 0}{\lim }\left(\frac{1}{\sin x}-\frac{1}{x}\right)$

Evaluate the following limit, if necessary use L’Hospital Rule :

$\underset{x\to 1^{+}}{\lim }\left(\frac{2}{x^2-1}-\frac{x}{x-1}\right)$

Evaluate the following limit, if necessary use L’Hospital Rule :

$\underset{x\to 0^{+}}{\lim }{x}^x$

Evaluate the following limit, if necessary use L’Hospital Rule :

$\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x$

Evaluate the following limit, if necessary use L’Hospital Rule :

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\sin x^{\tan x}$

Evaluate the following limit, if necessary use L’Hospital Rule :

If an initial amount $A_0$ of money is invested at an interest rate $r$ compounded $n$ Times a year, the value of the investment after $t$ Years is $A=A_0{\left(1+\frac{r}{n}\right)}^{nt}$. If the interest is compounded continuously, (that is as $n➝\infty$), show that the amount after $t$ Years is $A=A_0{e}^{rt}$.