(i) If f(x)={x−|x|xif x≠02if x=0, show that limx→ 0 f(x) does no

1.	(i) If f(x)={x−\|x\|xif x≠02if x=0, show that limx→ 0 f(x) does not exist. (ii) Evaluate limx→ 0 sin x−2 xin 3x+sin 5xx. Or (i) Find the derivative of (x−1)(x−2)(x−3)(x−4). (ii) Differentiate xex by using first principle.
Answer» (i) If $f (x) = {\begin{matrix} \frac{x - \| x \|}{x} & i f x \neq 0 2 & i f x = 0 \end{matrix}$ , show that $l i m x \to 0 f (x)$ does not exist. (ii) Evaluate $l i m x \to 0 \frac{s i n x - 2 x i n 3 x + s i n 5 x}{x} .$ Or (i) Find the derivative of $\frac{(x - 1) (x - 2)}{(x - 3) (x - 4)} .$ (ii) Differentiate $x e^{x}$ by using first principle.

(i) If f(x)={x−|x|xif x≠02if x=0, show that limx→ 0 f(x) does not exist. (ii) Evaluate limx→ 0 sin x−2 xin 3x+sin 5xx. Or (i) Find the derivative of (x−1)(x−2)(x−3)(x−4). (ii) Differentiate xex by using first principle.

Answer»

(i) If $f (x) = {\begin{matrix} \frac{x - | x |}{x} & i f x \neq 0 2 & i f x = 0 \end{matrix}$ , show that $l i m x \to 0 f (x)$ does not exist.

(ii) Evaluate $l i m x \to 0 \frac{s i n x - 2 x i n 3 x + s i n 5 x}{x} .$

(i) Find the derivative of $\frac{(x - 1) (x - 2)}{(x - 3) (x - 4)} .$

(ii) Differentiate $x e^{x}$ by using first principle.