_ x 3 [ 5^ x-3 -4^ x-3 (x-3) ]= - Vidyadip

MCQ

$
\lim _{x \rightarrow 3}\left[\frac{5^{x-3}-4^{x-3}}{\sin (x-3)}\right]=
$

A
$\log 5-4$
✓
$\log \frac{5}{4}$
C
$\frac{\log 5}{\log 4}$
D
$\frac{\log 5}{4}$

✓

Answer

Correct option: B.

$\log \frac{5}{4}$

(B) $\log \frac{5}{4}$
Hint:
$
\begin{aligned}
& \lim _{x \rightarrow 3} \frac{5^{x-3}-4^{x-3}}{\sin (x-3)} \\
& \text { put } x-3=h \\
\therefore \quad & x=3+h \\
& \text { As } x \rightarrow 3, h \rightarrow 0 \\
\therefore \quad & \text { Required limit } \\
& =\lim _{h \rightarrow 0} \frac{5^h-4^h}{\sinh }\\
& =\lim _{h \rightarrow 0} \frac{\left(5^h-1\right)-\left(4^h-1\right)}{\sin h} \\
& =\frac{\lim _{h \rightarrow 0} \frac{\left(5^h-1\right)}{h}-\frac{\left(4^h-1\right)}{h}}{\lim _{h \rightarrow 0} \frac{\sin h}{h}} \quad \cdots\left[\lim _{x \rightarrow 0} \frac{a^x-1}{x}=\log a\right] \\
& =\frac{\log 5-\log 4}{1} \\
= & \log \left(\frac{5}{4}\right)
\end{aligned}
$

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

The area of an equilateral triangle inscribed in the circle $x^2 + y^2 - 6x - 8y - 25 = 0$ is:

If $(1+i)(1+2 i)(1+3 i) \ldots .(1+n i)=a+i b$, then $2.5 .10 \ldots\left(1+n^2\right)$ is equal to

A and B are two events such that $P ( A )=0.8$, $P ( B )=0.6$ and $P ( A \cap B )=0.5$, then the value of $P ( A / B )$ is

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

$\lim _{x \rightarrow 0}\left[\frac{\log (5+x)-\log (5-x)}{\sin x}\right]=$

Maximum and minimum values of $\sin ^4 \theta+\cos ^4 \theta$ are

$\lim _{x \rightarrow 1} \frac{x^8-2 x+1}{x^4-2 x+1}$ equals

$8\sin\frac{\text{x}}{8}\cos\frac{\text{x}}{2}\cos\frac{\text{x}}{4}\cos\frac{\text{x}}{8}$ is equal to :

The value of $\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin 359^{\circ}$ is equal to

In a class of $175$ students the following data shows the number of students opting one or more subjects. Mathematics $100$; Physics $70$; Chemistry $40$; Mathematics and Physics $30$; Mathematics and Chemistry $28$; Physics and Chemistry $23$; Mathematics, Physics and Chemistry $18$. How many students have offered Mathematics alone?