Question

Limit x tends to zero
Sin x^2/x sinx
Answer is 1

Answer 1

$\rm \longrightarrow \displaystyle \large\lim_ { \rm \: x \to \:0 }{ \rm \: \dfrac{sin \: {x}^{2} }{ x \: sin \: x} }$

we know that :-

$\rm \boxed{ \displaystyle \large\lim_ { \rm \: x \to \:y }{ \rm \: \dfrac{sin \: y }{ y} = 1 }}$

$\rm \longrightarrow \displaystyle \large\lim_ { \rm \: x \to \:0 }{ \rm \: \dfrac{ \dfrac{sin \: {x}^{2}}{ {x}^{2} } \times {x}^{2} }{ x \times \dfrac{sin \: x}{x} \times x} }$

$\rm \longrightarrow \displaystyle \large\lim_ { \rm \: x \to \:0 }{ \rm \: \dfrac{ 1\times {x}^{2} }{ x \times 1 \times x} }$

$\rm \longrightarrow \displaystyle \large\lim_ { \rm \: x \to \:0 }{ \rm \: \dfrac{ {x}^{2} }{ {x}^{2} } }$

$\rm \longrightarrow 1$

Answer : 1

Additional Information :-

d(e^x)/dx = e^x

d(x^n)/dx = n x^(n-1)

d(ln x)/dx = 1/x

d(sin x)/dx = cos x

d(cos x)/dx = - sin x

d(tan x)/dx = sec² x

d(sec x)/dx = tan x * sec x

d(cot x)/dx = - cosec²x

d(cosec x)/dx = - cosec x * cot x