_ x 1 2^ 2 x-2 -2^x+1 ^2(x-1) = - Vidyadip

MCQ

$\lim _{x \rightarrow 1} \frac{2^{2 x-2}-2^x+1}{\sin ^2(x-1)}=$

A
$2 \log 2$
B
$2(\log 2)^2$
C
$\frac{1}{2}(\log 2)^2$
✓
$(\log 2)^2$

✓

Answer

Correct option: D.

$(\log 2)^2$

(d) : We have, $\lim _{x \rightarrow 1} \frac{2^{2 x-2}-2^x+1}{\sin ^2(x-1)}$
Given limit is of the form $\frac{0}{0}$. So, applying L' Hopital's rule, we get
$
\lim _{x \rightarrow 1} \frac{2^{2 x-2} \log 2(2)-2^x \log 2+0}{2 \sin (x-1) \cos (x-1)}
$
$
=\lim _{x \rightarrow 1} \frac{2^{2 x-2} 2 \times(\log 2)-2^x \log 2}{\sin 2(x-1)}
$ $\left(\frac{0}{0} \text { form }\right)
$
Again applying L'Hopital's Rule,
$
\lim _{x \rightarrow 1} \frac{2 \log 2 \times 2^{2 x-2}(2 \log 2)-2^x(\log 2)^2}{2 \cos 2(x-1)}
$
$
=\frac{4(\log 2)^2-2(\log 2)^2}{2}=\frac{2(\log 2)^2}{2}=(\log 2)^2
$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Limits→Limits — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $f (x)=\frac{2 x-1}{x+5}(x \neq 5)$, then $f ^{-1}(x)$ is equal to

If $x=3+ i$, then $x^3-3 x^2-8 x+15=$

If $f(x)$ is continuous in $[-2,2]$. where$f(x)=\left\{\begin{array}{ll}x+a, & x<0 \\x, & 0 \leq x<1, \text { then } \\b-x, & x \geq 1\end{array}\right.$

The number of lines that are parallel to 2x + 6y + 7 = 0 and have an intercept of length 10 between the co-ordinate axes is

If $\theta=\frac{\pi}{2^n+1}$, then
$\cos \theta \cos 2 \theta \cos 2^2 \theta \ldots \cos 2^{n-1} \theta$ is equal to

The distance between the foci of an ellipse is 16 and eccentricity is $\frac{1}{2}$ . Length of the major axis of the ellipse is

If $z$ and $\omega$ are two non-zero complex numbers such that $|z|=|\omega|$ and $\operatorname{Arg}(z)+\operatorname{Arg}(\omega)=\pi$, then $z =$

If the origin is shifted to the point (-3, 2), axes remaining parallel, the new co-ordinate of the point is (3, 1), then its old co-ordinate will be

A man running round a race-course notes that the sum of the distance of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 metres. The area of the path he encloses in square metres is

If for a G.P. $\frac{t_6}{t_3}=\frac{1458}{54}$ then $r=$ ?