Evaluate the following- - lim x - 1 x 2 - 3 x - 2 x 2 - 4 x - 3- lim x - 2 x 3 - x 2 - 12 x 3 - x 2 - x - 2- lim x - 2 x 2 - x - 2 - 4 x 2 - 3 x - 2 - 4- lim x - 3 x 3 - 3 x 2 - x - 3 x 3 - 6 x 2 - 9 x -

Use L Hospital rule lim_x→ 1x^2 3x+2x^2 4x+3=lim_x→ 12x 32x 4=2(1) 32(1) 4= 12 lim_x→ 2x^3+x^2 12x^3 x^2 x 2=lim_x→ 23x^2+2x3x^2 2x 1=3(4)+2(2)3(4) 2(2 ...

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Byju's Answer

Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

Evaluate the ...

Question

Evaluate the following:
$\begin{matrix} {lim}_{x \to 1} \frac{x^{2} - 3 x + 2}{x^{2} - 4 x + 3} {lim}_{x \to 2} \frac{x^{3} + x^{2} - 12}{x^{3} - x^{2} - x - 2} {lim}_{x \to \sqrt{2}} \frac{x^{2} + x \sqrt{2} - 4}{x^{2} - 3 x \sqrt{2} + 4} {lim}_{x \to 3} \frac{x^{3} - 3 x^{2} - x + 3}{x^{3} - 6 x^{2} + 9 x} \end{matrix}$

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Solution

Use L-Hospital rule
$lim x \to 1 \frac{x^{2} - 3 x + 2}{x^{2} - 4 x + 3} = lim x \to 1 \frac{2 x - 3}{2 x - 4} = \frac{2 (1) - 3}{2 (1) - 4} = \frac{- 1}{2}$

$lim x \to 2 \frac{x^{3} + x^{2} - 12}{x^{3} - x^{2} - x - 2} = lim x \to 2 \frac{3 x^{2} + 2 x}{3 x^{2} - 2 x - 1} = \frac{3 (4) + 2 (2)}{3 (4) - 2 (2) - 1} = \frac{16}{7}$

$lim x \to \sqrt{2} \frac{x^{2} + x \sqrt{2} - 4}{x^{2} - 3 x \sqrt{2} + 4} = lim x \to \sqrt{2} \frac{2 x + \sqrt{2}}{2 x - 3 \sqrt{2}} = \frac{2 (\sqrt{2}) + \sqrt{2}}{2 (\sqrt{2}) - (3 \sqrt{2})} = - 3$

$lim x \to 3 \frac{x^{3} - 3 x^{2} - x + 3}{x^{3} - 6 x^{2} + 9 x} = lim x \to 3 \frac{3 x^{2} - 6 x - 1}{3 x^{2} - 12 x + 9} = \frac{6 x - 6}{6 x - 12} = 2$ .

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Similar questions

\lim_{x \to - 2} \frac{x^{3} + x^{2} + 4 x + 12}{x^{3} - 3 x + 2}

Q. Find the limits :

i .

lim x \to 1 [\frac{x^{2} + 1}{x + 100}]

i i .

lim x \to 2 [\frac{x^{3} - 4 x^{2} + 4 x}{x^{2} - 4}]

i i i .

lim x \to 2 [\frac{x^{2} - 4}{x^{3} - 4 x^{2} + 4 x}]

i v .

lim x \to 2 [\frac{x^{3} - 2 x^{2}}{x^{2} - 5 x + 6}]

v .

lim x \to 1 [\frac{x - 2}{x^{2} - x} - \frac{1}{x^{3} - 3 x^{2} + 2 x}]

Q. Evaluate:
1.

{lim}_{x \to 2} \frac{x^{3} - 3 x^{2} + 4}{x^{4} - 8 x^{2} + 16}

{lim}_{x \to 1} [\frac{2}{1 - x^{2}} + \frac{1}{x - 1}]

{lim}_{x \to - 3} [\frac{1}{x^{2} + 4 x + 3} + \frac{1}{x^{2} + 8 x + 15}]

Q. Evaluate the following one sided limits:

(i)

\lim_{x \to 2^{+}} \frac{x - 3}{x^{2} - 4}

(ii)

\lim_{x \to 2^{-}} \frac{x - 3}{x^{2} - 4}

(iii)

\lim_{x \to 0^{+}} \frac{1}{3 x}

(iv)

\lim_{x \to - 8^{+}} \frac{2 x}{x + 8}

(v)

\lim_{x \to 0^{+}} \frac{2}{x^{1 / 5}}

(vi)

\lim_{x \to \frac{π}{2}} \tan x

(vii)

\lim_{x \to - π / 2^{+}} \sec x

(viii)

\lim_{x \to 0^{-}} \frac{x^{2} - 3 x + 2}{x^{3} - 2 x^{2}}

(ix)

\lim_{x \to - 2^{+}} \frac{x^{2} - 1}{2 x + 4}

(x)

\lim_{x \to 0^{-}} 2 - \cot x

(xi)

\lim_{x \to 0^{-}} 1 + cosec x

Evaluate the following one sided limits:

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

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

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

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

$(v) {lim}_{x \to 0^{+}} \frac{2}{x^{\frac{1}{5}}}$

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

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

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

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

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

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

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