Question

#challenge_question_for_math_lover

$\rm{ \displaystyle \lim _{ \rm \: x \to1} \frac{ \int_{0} ^{( \rm \: x - 1) ^{2} } \rm t \cos tdt}{ \rm(x - 1) \sin(x - 1)} }$
Find the value ​

Answer 1

$\implies\rm{ \displaystyle\lim _{ \rm x \to1} \dfrac{\displaystyle \int_{0}^{( \rm x - 1) ^{2} } \rm t \cos tdt}{ \rm(x - 1) \sin(x - 1)} }$

Now if we put x = 1 then we will get 0/0 form.

So , we will apply L' Hospital rule and differentiate both numerator and denominator seperately.

$\implies\rm{ \displaystyle\lim _{ \rm x \to1} \dfrac{ \dfrac{d \bigg( \displaystyle \int_{0}^{( \rm x - 1) ^{2} } \rm t \cos tdt \bigg)}{dx} }{ \frac{d \bigg(\rm(x - 1) \sin(x - 1)\bigg )}{dx} } }$

Now to differentiate the integral function we will apply Leibnitz rule. [To understand Leibnitz refer attachment. ]

$\rm\implies{ \displaystyle\lim _{ \rm x \to1} \dfrac{ \rm{(x - 1)}^{2} \: \times cos(x - 1)^{2} \times \dfrac{d (x - 1)^{2}}{dx} }{ \rm(x - 1) cos(x - 1) + sin(x - 1)} }$

$\rm\implies{ \displaystyle\lim _{ \rm x \to1} \dfrac{ 2 \rm{(x - 1)}^{2} \: \times cos(x - 1)^{2} \times {(x - 1)}}{ \rm(x - 1) cos(x - 1) + sin(x - 1)} }$

$\rm\implies{ \displaystyle\lim _{ \rm x \to1} \dfrac{ 2\rm{(x - 1)}^{3} \: \times cos(x - 1)^{2} }{ \rm(x - 1) cos(x - 1) + sin(x - 1)} }$

Now again if we put x = 1 in the above expression, then we will get 0/0 form.

So , now we will further simplify the above expression by dividing (x-1) from numerator and denominator.

$\rm\implies{ \displaystyle\lim _{ \rm x \to1} \dfrac{ 2 \bigg [\rm{(x - 1)}^{3} \: \times cos(x - 1)^{2} \bigg] \bigg/(x - 1)}{\bigg [ \rm(x - 1) cos(x - 1) + sin(x - 1)\bigg] \bigg/(x - 1)} }$

$\rm\implies{ \displaystyle\lim _{ \rm x \to1} \dfrac{ 2\rm{(x - 1)}^{2} \: \times cos(x - 1)^{2} }{ \rm cos(x - 1) + \dfrac{sin(x - 1)}{(x - 1)} } }$

Now put x = 1

$\rm\implies{\dfrac{ 2\rm{(1 - 1)}^{2} \: \times cos(1 - 1)^{2} }{ \rm cos(1- 1) + 1 } } = \dfrac{2 \times 0 \times 1}{1 + 1}$

$\rm\implies \dfrac{2 \times 0 \times 1}{1 + 1} = \dfrac{0}{2}$

$\rm\implies \dfrac{0}{2} = 0$

Answer : 0

Answer 2

$\huge\underline{\bf \orange{Aηsωeя :}}$

Hope it helps...