Limits and Derivatives

15. $\underset{x\to \pi }{lim}\frac{sin(\pi -x)}{\pi (\pi -x)}=\frac{1}{\pi }*\underset{x\to \pi }{lim}\frac{sin(\pi -x)}{\pi -x}$

$=\frac{1}{\pi }.$

14. $\underset{x\to 0}{lim}\frac{sinax}{sinbx}=\underset{x\to 0}{lim}\frac{\frac{sinax}{ax}*ax}{\frac{sinbx}{bx}*bx}$

$=\frac{a}{b}$

13. $\underset{x\to 0}{lim}\frac{sinax}{bx}=\frac{1}{b}\underset{x\to 0}{lim}\frac{sinax}{ax}*a$

$=\frac{a}{b}\underset{x\to 0}{lim}\frac{sinax}{ax}$ 

$=\frac{a}{b}$

Kindly go through the solution

$=\frac{1}{2*(-2)}=-\frac{1}{4}.$

11. $\underset{x\to 1}{lim}\frac{a{x}^2+bx+c}{c{x}^2+bx+a}=\frac{a+b+c}{c+b+a}=1$

10.$=\underset{z\to 1}{lim}\left[\frac{z^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{z-1}\right]÷\underset{z\to 1}{lim}\left[\frac{z^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}-1^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{z-1}\right]$

$=\underset{z\to 1}{lim}\left[\frac{z^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{z-1}\right]÷\underset{z\to 1}{lim}\left[\frac{z^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}-1^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{z-1}\right]$

We know that,

$\underset{}{}$$\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

So,

$=\frac{1}{3}*6=2$

Kindly go through the solution

8.

$=\underset{x\to 3}{lim}\frac{(x+3)(x^2+9)}{2x+1.}$

$=\frac{(3+3)\left(3^2+9\right)}{2*3+1}$

$=\frac{6*18}{6+1}$

$=\frac{108}{7}$

7.

6. $\underset{x\to 0}{lim}\frac{{(x+1)}^5-1}{x}=\underset{x\to 0}{lim}\frac{x^5+5x^4+10x^3+10x^2+5x+1-1}{}$

Q6. $\underset{x\to 0}{\lim }\frac{{\left(x+1\right)}^5-1}{x}$

6. $\underset{x\to 0}{lim}\frac{{(x+1)}^5-1}{x}=\underset{x\to 0}{lim}\frac{x^5+5x^4+10x^3+10x^2+5x+1-1}{}$ $=\underset{x\to 0}{lim}{x}^4+5x^3+10x^2+10x+5.$ (0)4 + 5(0)3 + 10(0)2 + 10(0) + 5 = 5.