Differentiation (Part-38) | S N De

 $\,10.\,$ Evaluate the following limits :

$\,(i)\,~~\lim_{x \to 2} \frac{x(x^2-4)}{x^2-5x+6}$ 

Sol. $\,~~~\lim_{x \to 2} \frac{x(x^2-4)}{x^2-5x+6}\\=\lim_{x \to 2}\frac{x(x^2-2^2)}{x^2-2x-3x+6}\\=\lim_{x \to 2}\frac{x(x+2)(x-2)}{x(x-2)-3(x-2)}\\=\lim_{x \to 2}\frac{x(x+2)(x-2)}{(x-2)(x-3)}\\=\lim_{x \to 2}\frac{x(x+2)}{x-3}~~~[\because x \to 2, \,x  \neq 2]\\=\frac{2(2+2)}{2-3}\\=-8\,\,\text{(ans.)}$

$(ii)~\lim_{h  \to 1}~\frac{h^2+2h-3}{h-1}$

Sol. $\,~\lim_{h  \to 1}~\frac{h^2+2h-3}{h-1}\\=\lim_{h  \to 1}\frac{h^2+3h-h-3}{h-1}\\=\lim_{h  \to 1}\frac{h(h+3)-1(h+3)}{h-1}\\=\lim_{h  \to 1}\frac{(h+3)(h-1)}{h-1}\\=\lim_{h  \to 1}~~(h+3)~~[\because h \to 1, \,h  \neq 1]\\=1+3\\=4\,\,\text{(ans.)}$

$\,(iii)\, \lim_{ x \to -\frac 12 }\frac{6x^2-7x-5}{2x^2-x-1}$

Sol. $\,\,\lim_{ x \to -\frac 12 } \frac{6x^2-7x-5}{2x^2-x-1}\\=\lim_{ x \to -\frac 12 }\frac{6x^2-10x+3x-5}{2x^2-2x+x-1}\\=\lim_{ x \to -\frac 12 }\frac{2x(3x-5)+1(3x-5)}{2x(x-1)+1(x-1)}\\=\lim_{ x \to -\frac 12 }\frac{(3x-5)(2x+1)}{(x-1)(2x+1)}\\=\frac{3x-5}{x-1}~~[\because x \to -\frac 12, \,(2x+1) \neq 0]\\=\frac{3 \times  (-1/2)-5}{-\frac 12-1}\\=\frac{-13/2}{-3/2}=\frac{13}{3}\,\,\text{(ans.)}$

$\,(iv)~~\lim_{u  \to -3}\,\left[\frac{1}{u+3}+\frac{6}{u^2-9}\right]$

Sol.$\,~~~~\lim_{u  \to -3}\,\left[\frac{1}{u+3}+\frac{6}{u^2-9}\right]\\=\lim_{u  \to -3}\left[\frac{1}{u+3}+\frac{6}{(u+3)(u-3)}\right]\\=\lim_{u  \to -3}\left[\frac{u-3+6}{(u+3)(u-3)}\right]\\=\lim_{u  \to -3}\frac{u+3}{(u+3)(u-3)}\\=\lim_{u  \to -3}\frac{1}{u-3}\\=\frac{1}{-3-3}\\=-\frac 16~~\text{(ans.)}$

$\,(v)\,~\lim_{x \to 1}~\frac{x^3-1}{x^2-1}$

Sol. $\,~~~\lim_{x \to 1}\frac{x^3-1}{x^2-1}\\=\lim_{x \to 1}\frac{(x-1)(x^2+x+1)}{(x-1)(x+1)}\\=\lim_{x \to 1}\frac{x^2+x+1}{x+1}~~[\because x \to 1, x \neq 1]\\=\frac{1^2+1+1}{1+1}\\=\frac 32\,\,\text{(ans.)}$

$\,(vi)\lim_{x \to 1}\,\frac{x^2+5x-6}{x^2-3x+2}$

Sol. $\,~~~\lim_{x \to 1}\,\frac{x^2+5x-6}{x^2-3x+2}\\=\lim_{x \to 1}\frac{x^2+6x-x-6}{x^2-2x-x+2}\\=\lim_{x \to 1}\frac{x(x+6)-1(x+6)}{x(x-2)-1(x-2)}\\=\lim_{x \to 1}\frac{(x+6)(x-1)}{(x-2)(x-1)}\\=\lim_{x \to 1}\left(\frac{x+6}{x-2}\right)~~~[\because x \to 1, x \neq 1]\\=\frac{1+6}{1-2}\\=\frac{7}{-1}\\=-7\,\,\text{(ans.)}$

$\,(vii)\,\lim_{x \to  (a/2)}\frac{8x^2-10ax+3a^2}{4x^2+4ax-3a^2}$

Sol. $\,\lim_{x \to  (a/2)}\frac{8x^2-10ax+3a^2}{4x^2+4ax-3a^2}\\=\lim_{x \to  (a/2)}\frac{8x^2-6ax-4ax+3a^2}{4x^2+6ax-2ax-3a^2}\\=\lim_{x \to  (a/2)}\frac{2x(4x-3a)-a(4x-3a)}{2x(2x+3a)-a(2x+3a)}\\=\lim_{x \to  (a/2)}\frac{(4x-3x)(2x-a)}{(2x+3a)(2x-a)}\\=\lim_{x \to  (a/2)}\frac{4x-3a}{2x+3a}~~~[\because x \to (a/2), x \neq \frac a2]\\=\frac{4 \times \frac a2-3a}{2 \times \frac a2+3a}\\=\frac{2a-3a}{a+3a}\\=\frac{-a}{4a}\\=-\frac 14\,\,\text{(ans.)}$

$\,(viii)\lim_{x \to  (1/2)}\,\frac{8x^3-1}{6x^2-5x+1}$

Sol. $\,\lim_{x \to  (1/2)}\,\frac{8x^3-1}{6x^2-5x+1}\\=\lim_{x \to  (1/2)}\frac{(2x)^3-1^3}{6x^2-3x-2x+1}\\=\lim_{x \to  (1/2)}\frac{(2x-1)[(2x)^2+2x \cdot 1+1^2]}{3x(2x-1)-1(2x-1)}\\=\lim_{x \to  (1/2)}\frac{(2x-1)(4x^2+2x+1)}{(2x-1)(3x-1)}\\=\lim_{x \to  (1/2)}\frac{4x^2+2x+1}{3x-1}\\~~~[\because  x \to \frac 12, x \neq \frac 12 \Rightarrow 2x-1 \neq 0]\\=\frac{4 \times \left(\frac 12\right)^2+ 2 \times \frac 12+1}{3 \times \frac 12-1}\\=\frac{1+1+1}{\frac 12}\\= 3 \times 2\\=6\,\,\text{(ans.)}$