Sint X + X COS X (L Hospital's rule...
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Sint X + X COS X (L Hospital's rule again). Illustration 4: Evaluate lim x-*y y * - y

JEE/Engineering Exams
Maths
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( L=lim _{x rightarrow y} frac{x^{y}-y^{x}}{x^{x}-y^{y}} ) for thin limit, yourve Put ( x=y+epsilon ), ( L=lim _{epsilon rightarrow 0} frac{(y+epsilon)^{y}-y^{y+epsilon}}{(y+epsilon)^{y+t}-y^{y}} ) ( left.y^{t}=e^{t(n y} uright)+t ln y ) ( begin{aligned} L &=lim _{t rightarrow 0} y^{y}left(1+frac{t}{y}right)^{y}-y^{y} y^{t} & y y yleft(1+frac{t}{y}right)^{y+epsilon}-y^{y} &=lim _{t rightarrow 0} frac{left(1+frac{epsilon}{y}right)^{y}-y^{t}}{y^{t}left(1+frac{epsilon}{y}right)^{y+epsilon}-1} end{aligned} ) ( u lim _{epsilon rightarrow 0} frac{(1+epsilon)-(1-epsilon ln y)}{(1+epsilon ln y)(1+epsilon)-1} ) ( =frac{1-ln y}{1+ln y} )
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