\mathop \lim \limits_x \to 2 3^x + 3^3 - x - 12 \over 3^ - x/2 - 3^1 - x is equal to \underline\hspace2cm .

Limits Continuity and Differentiability PYQ — JEE Main 2020 · 7 Jan · Shift 1 · Q30 with Solution

View written solutionFree

Correct answer: 36

We need to evaluate

L=\lim_{x\to 2}\frac{3^x+3^{3-x}-12}{3^{-x/2}-3^{1-x}}.

First check the form at $x=2$ :

Numerator:

3^2+3^{3-2}-12=9+3-12=0.

Denominator:

3^{-2/2}-3^{1-2}=3^{-1}-3^{-1}=0.

So this is of the indeterminate form $\frac{0}{0}$ , hence we can use L'Hospital's Rule.

Differentiate numerator and denominator with respect to $x$ .

For the numerator,

N(x)=3^x+3^{3-x}-12.

Then

N'(x)=3^x\ln 3+3^{3-x}(\ln 3)(-1) =\ln 3\,(3^x-3^{3-x}).

For the denominator,

D(x)=3^{-x/2}-3^{1-x}.

Then

\frac{d}{dx}\left(3^{-x/2}\right)=3^{-x/2}\ln 3\left(-\frac12\right),

and

\frac{d}{dx}\left(-3^{1-x}\right)=-\left(3^{1-x}\ln 3(-1)\right)=3^{1-x}\ln 3.

D'(x)= -\frac12 3^{-x/2}\ln 3 + 3^{1-x}\ln 3 =\ln 3\left(3^{1-x}-\frac12 3^{-x/2}\right).

Apply L'Hospital's Rule:

L=\lim_{x\to 2}\frac{N'(x)}{D'(x)} =\lim_{x\to 2}\frac{\ln 3\,(3^x-3^{3-x})}{\ln 3\left(3^{1-x}-\frac12 3^{-x/2}\right)}.

Cancel $\ln 3$ :

L=\lim_{x\to 2}\frac{3^x-3^{3-x}}{3^{1-x}-\frac12 3^{-x/2}}.

Substitute $x=2$ now:

Numerator:

3^2-3^{1}=9-3=6.

Denominator:

3^{1-2}-\frac12 3^{-2/2}=3^{-1}-\frac12 3^{-1} =\frac13-\frac16=\frac16.

Thus,

L=\frac{6}{1/6}=36.

Therefore, the required integer is

\boxed{36}.

More from Limits Continuity and Differentiability

Let S be the set of points where the function, ƒ(x) = |2-|x-3||, x $\in$ R is not differentiable. Then $x \in S \sum f (f (x))$ is equal to .2020 · Numerical
If the function ƒ defined on $(- \frac{1}{3}, \frac{1}{3})$ by f(x) = ${\frac{1}{x} lo g_{e} (\frac{1 + 3 x}{1 - 2 x}), k, w h e n x e 0 w h e n x = 0$ …2020 · Numerical
$x \to 0 lim (\frac{3 x ^{2} + 2}{7 x ^{2} + 2})^{\frac{1}{x ^{2}}}$ is equal to2020 · MCQ
Let S be the set of all functions ƒ : [0,1] $\to$ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $\in$ (0,1), depending on ƒ, such that2020 · MCQ
Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $\in$ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for any c $\in$ (a, b), $\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )}$ is greater than :2020 · MCQ
If $f (x) = ⎩ ⎨ ⎧ \frac{s i n ( a + 2 ) x + s i n x}{x}; b; \frac{( x + 3 x ^{2} ) ^{\frac{1}{3}} - x ^{\frac{1}{3}}}{x ^{\frac{4}{3}}}; x < 0 x = 0 x > 0$ …2020 · MCQ
Let [t] denote the greatest integer $\leq$ t and $x \to 0 lim x [\frac{4}{x}] = A$ . Then the function, f(x) = [x2]sin( $π$ x) is discontinuous, when x is equal to :2020 · MCQ
$x \to 0 lim \frac{s i n ^{2} x}{2 - 1 + c o s x}$ equals:2019 · MCQ

View written solutionFree

Correct answer: 36

We need to evaluate

L=\lim_{x\to 2}\frac{3^x+3^{3-x}-12}{3^{-x/2}-3^{1-x}}.

First check the form at $x=2$ :

Numerator:

3^2+3^{3-2}-12=9+3-12=0.

Denominator:

3^{-2/2}-3^{1-2}=3^{-1}-3^{-1}=0.

So this is of the indeterminate form $\frac{0}{0}$ , hence we can use L'Hospital's Rule.

Differentiate numerator and denominator with respect to $x$ .

For the numerator,

N(x)=3^x+3^{3-x}-12.

Then

N'(x)=3^x\ln 3+3^{3-x}(\ln 3)(-1) =\ln 3\,(3^x-3^{3-x}).

For the denominator,

D(x)=3^{-x/2}-3^{1-x}.

Then

\frac{d}{dx}\left(3^{-x/2}\right)=3^{-x/2}\ln 3\left(-\frac12\right),

and

\frac{d}{dx}\left(-3^{1-x}\right)=-\left(3^{1-x}\ln 3(-1)\right)=3^{1-x}\ln 3.

D'(x)= -\frac12 3^{-x/2}\ln 3 + 3^{1-x}\ln 3 =\ln 3\left(3^{1-x}-\frac12 3^{-x/2}\right).

Apply L'Hospital's Rule:

L=\lim_{x\to 2}\frac{N'(x)}{D'(x)} =\lim_{x\to 2}\frac{\ln 3\,(3^x-3^{3-x})}{\ln 3\left(3^{1-x}-\frac12 3^{-x/2}\right)}.

Cancel $\ln 3$ :

L=\lim_{x\to 2}\frac{3^x-3^{3-x}}{3^{1-x}-\frac12 3^{-x/2}}.

Substitute $x=2$ now:

Numerator:

3^2-3^{1}=9-3=6.

Denominator:

3^{1-2}-\frac12 3^{-2/2}=3^{-1}-\frac12 3^{-1} =\frac13-\frac16=\frac16.

Thus,

L=\frac{6}{1/6}=36.

Therefore, the required integer is

\boxed{36}.

Limits Continuity and Differentiability question

Limits Continuity and Differentiability question

More from Limits Continuity and Differentiability

Limits Continuity and Differentiability question

Limits Continuity and Differentiability question

More from Limits Continuity and Differentiability