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Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

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\[\mathop {\lim }\limits_{x \to 0} \frac{{{x^2}y({e^{xy}} - 1 - xy)}}{{{{({x^2} + {y^2})}^2}}}\]

A particle moves in the xy-plane along the parabolic trajectory $y=x^2$. As the particle approaches the origin, evaluate $L=\mathop {\lim }\limits_{x \to 0} \frac{{{x^2}y({e^{xy}} - 1 - xy)}}{{{{({x^2} + {y^2})}^2}}}$.

Solution

Substituting $y=x^2$, we have $L=\mathop {\lim }\limits_{x \to 0} \frac{{{x^4}({e^{x^3}} - 1 - x^3)}}{{{{({x^2} + {x^4})}^2}}}$

Or, $L=\mathop {\lim }\limits_{x \to 0} \frac{{{x^4}({e^{x^3}} - 1 - x^3)}}{x^4{{{(1 + {x^2})}^2}}}=\mathop {\lim }\limits_{x \to 0} \frac{{{e^{x^3}} - 1 - x^3}}{{{{(1 + {x^2})}^2}}}$

It may seem like exponential series expansion formula should be used, but it is not required. Direct substitution of x = 0 yields L = 0.