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Limits — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

\displaystyle\lim_{\substack{x\to 0\\ y\to 0}}\dfrac{y^{2}+\sin x}{x^{2}+\sin y^{2}}

when

(x,y)\to(0,0)

along the curve

x=y^{2}

0

1

2

correct

DDoes not exist

Solution

Step 1: Substitute

x=y^{2}

. The expression becomes:

\dfrac{y^{2}+\sin y^{2}}{y^{4}+\sin y^{2}}

Step 2: As

y\to 0

\sin y^{2}\sim y^{2}

. Substitute:

\dfrac{y^{2}+y^{2}}{y^{4}+y^{2}}=\dfrac{2y^{2}}{y^{2}(y^{2}+1)}=\dfrac{2}{y^{2}+1}

Step 3: As

y\to 0

\dfrac{2}{0+1}=2

Correct answer: (3)

2

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