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  • Need solution for RD Sharma maths class 12 chapter 8 Continuity exercise multiple choice question 7

Need solution for RD Sharma maths class 12 chapter 8 Continuity exercise multiple choice question 7

Answers (1)

Answer:

 The correct option is (C)

Hint:

 Use the given formula:

\text { (i) } \lim _{x \rightarrow 0} \frac{\log (1-x)}{x}=1 \text { and } \lim _{x \rightarrow 0} \frac{\tan x}{x}=1

(ii)  A function f(x) is said to be continuous at a point x = a of its domain

\begin{aligned} &\lim _{x \rightarrow a^{+}} f(a+h)=\lim _{x \rightarrow a^{-}} f(a-h)=f(a) \\ &(\text {iii}) \lim _{x \rightarrow a}\{f(x) g(x)\}=1 m, \text { where } \lim _{x \rightarrow a} f(x)=1, \lim _{x \rightarrow a} g(x)=m \end{aligned}

Given:

 f(x)=(x+1)^{\cot x}

Solution:

f(x)=(x+1)^{\cot x}

taking log on both sides

\log f(x)=(\cot x)(\log (x+1))

\lim _{x \rightarrow 0} \log f(x)=\lim _{x \rightarrow 0}\left(\frac{\frac{\log (x+1)}{x}}{\frac{\tan x}{x}}\right)

Using formula (iii)

\lim _{x \rightarrow 0} \log f(x)=\frac{\lim _{x \rightarrow 0} \frac{\log (x+1)}{x}}{\lim _{x \rightarrow 0} \frac{\tan x}{x}}

Using standard limit formula (i)

\begin{aligned} &\lim _{x \rightarrow 0} f(x)=e \\ &f(0)=e \end{aligned}

So, the correct option is (C)

Posted by

Gurleen Kaur

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