Mathematics

# If $a,b$ and $c$ are real numbers then the value of $\mathop {\lim }\limits_{t \to 0} {l_n}\left( {\frac{1}{t}\int_0^1 {{{\left( {1 + a\sin bx} \right)}^{\frac{c}{x}}}dx} } \right)$ equals

$abc$

##### SOLUTION
Let $L={\lim}_{t\rightarrow\,0}\ln{\left(\dfrac{1}{t}\displaystyle\int_{0}^{t}{{\left(1+a\sin{bx}\right)}^{\frac{c}{x}}dx}\right)}$
It is of $\dfrac{0}{0}$ form.
Hence we use L'Hospitals's rule
$L={\lim}_{t\rightarrow\,0}\dfrac{{\left(1+a\sin{bt}\right)}^{\frac{c}{t}}\times 1}{1}$
$L={\lim}_{t\rightarrow\,0}\ln{{e}^{a\sin{bt}\times\dfrac{c}{t}}}$
$L={\lim}_{t\rightarrow\,0}\dfrac{ac\sin{bt}}{t}$ is of $\dfrac{0}{0}$ form.
$L={\lim}_{t\rightarrow\,0}\dfrac{ac\cos{bt}}{1}$
$=abc$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium
The value of $\int_{0}^{1} \dfrac {8\log (1 + x)}{1 + x^{2}} dx$ is
• A. $\dfrac {\pi}{2}\log 2$
• B. $2\pi\log 2$
• C. None of these
• D. $\pi\log 2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\displaystyle \int_{0}^{1}x^{n}\left ( \log x \right )^{m} dx$ $\ \forall m \epsilon I$ is ?
• A. $\displaystyle \frac{m! }{\left ( n + 1 \right )^{m}}$
• B. $\displaystyle \frac{m! }{\left ( n + 1 \right )^{m +1}}$
• C. None of the above
• D. $\displaystyle \frac{\left ( -1 \right )^{m}m! }{\left ( n+1 \right )^{m+1 }}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
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• A. $\displaystyle \tan^{-1} 1$
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1 Verified Answer | Published on 17th 09, 2020

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