(i) Find the derivative of cosec x from the first principle.

(ii) Evaluate $l i m_{x \to \sqrt{2}} \frac{x^{4} - 4}{x^{2} + 3 \sqrt{2} x - 8}$

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Solution

(i) - cosec x cot x (ii) $\frac{8}{5}$

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Similar questions

(i) Evaluate ${lim}_{x \to 1} \frac{x + x^{2} + x^{3} + . . . + x^{n} - n}{x - 1}$

(ii) Find the derivative $\sqrt{sin x}$ from first principle.

lim x \to \sqrt{2} \frac{x^{4} - 4}{x^{2} + 3 x \sqrt{2} - 8}

(i) Find the derivative of $\sqrt{\frac{(x - 1) (x - 2)}{(x - 3) (x - 4)}}$ + $\frac{s i n x}{1 + t a n x}$

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Evaluate the following one sided limits:

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$(i i) {lim}_{x \to 2^{-}} \frac{x - 3}{x^{2} - 4}$

$(i i i) {lim}_{x \to 0^{+}} \frac{1}{3 x}$

$(i v) {lim}_{x \to 8^{+}} \frac{2 x}{x + 8}$

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$(v i) {lim}_{x \to \frac{π^{-}}{2}} t a n x$

$(v i i) {lim}_{x \to \frac{π}{2^{+}}} s e c x$

$(v i i i) {lim}_{x \to 0^{-}} \frac{x^{2} - 3 x + 2}{x^{3} - 2 x^{2}}$

$(i x) {lim}_{x \to - 2^{+}} \frac{x^{2} - 1}{2 x + 4}$

$(x) {lim}_{x \to 0^{+}} (2 - c o t x)$

$(x i) {lim}_{x \to 0^{-}} 1 + c o s e c x$

(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.

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