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$$\lim_{(x,y)\to(0,0)}\frac{x^2+\sin^2y}{2x^2+y}$$

Above problem is in my textbook. it's different from others because I don't know how to process with trigometric element: in this example is $\sin^2y$

Thanks :)

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1 Answer

up vote 4 down vote accepted

HINT: Don’t let the trig function throw you. What happens if you approach the origin along the $x$-axis, with $y=0$ What happens if you approach the origin along the $y$-axis, with $x=0$? (You could also see what happens when you approach along the line $y=x$.)

By the way, a useful thing to remember is that when $x$ is very close to $0$, both $\sin x$ and $\tan x$ are very very close to $x$. Thus, to get a quick idea of what’s probably going on near the origin with this function, temporarily replace $\sin^2 y$ by $y^2$: it’s a good approximation when $y$ is close to $0$.

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