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 ...

1425

You visited us 1425 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

Evaluate the ...

Question

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

Open in App

Solution

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

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

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

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

Suggest Corrections

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:

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

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

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

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

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

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

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

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

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

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

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

Join BYJU'S Learning Program

Evaluate the following:limx→1x2−3x+2x2−4x+3limx→2x3+x2−12x3−x2−x−2limx→√2x2+x√2−4x2−3x√2+4limx→3x3−3x2−x+3x3−6x2+9x