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lim_((x,y)rarr(0,0))(y^(2)sen^(2)x)/(x^(4)+y^(4))

$11$ . $\lim _{(x, y) \rightarrow(0,0)} \frac{y^{2} \operatorname{sen}^{2} x}{x^{4}+y^{4}}$

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Find derivatives of using multiple formulae

Full solution

Q.

11

.

\lim _{(x, y) \rightarrow(0,0)} \frac{y^{2} \operatorname{sen}^{2} x}{x^{4}+y^{4}}

Analyze Behavior: Step $1$ : Analyze the behavior of the numerator and denominator near the point $(0,0)$ . $\newline$ Numerator: $y^2 \sin^2(x)$ approaches $0$ as both $y$ and $x$ approach $0$ . $\newline$ Denominator: $x^4 + y^4$ also approaches $0$ as $x$ and $y$ approach $0$ .
Apply Limit: Step $2$ : Apply the limit to the function. $\newline$ $\lim_{(x,y)\rightarrow(0,0)} \frac{y^2 \sin^2(x)}{x^4 + y^4}$ . $\newline$ Since both the numerator and denominator approach $0$ , we consider the rate at which they approach $0$ .
Use Polar Coordinates: Step $3$ : Use polar coordinates to simplify the limit. Let $x = r\cos(\theta)$ and $y = r\sin(\theta)$ , where $r$ is the radius and $\theta$ is the angle. Then, $x^4 + y^4 = r^4\cos^4(\theta) + r^4\sin^4(\theta) = r^4(\cos^4(\theta) + \sin^4(\theta))$ . $\sin^2(x) = \sin^2(r\cos(\theta))$ .
Substitute and Simplify: Step $4$ : Substitute polar coordinates into the limit. $\newline$ $\lim_{r\to 0} \left(\frac{r^2\sin^2(\theta) \cdot \sin^2(r\cos(\theta))}{r^4(\cos^4(\theta) + \sin^4(\theta))}\right)$ . $\newline$ This simplifies to $\frac{\sin^2(\theta) \cdot \sin^2(r\cos(\theta))}{r^2(\cos^4(\theta) + \sin^4(\theta))}$ .
Evaluate Limit: Step $5$ : Evaluate the simplified limit. $\newline$ As $r$ approaches $0$ , $\sin^2(r\cos(\theta))$ approaches $0$ faster than $r^2$ , making the whole expression approach $0$ .

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