What is the limit of #lim_((x,y)-&gt;(0,0))x^2/(x^2+y^2)#?

In order for this limit to exist, the fraction x2x2+y2 must approach the same value L, regardless of the path along which we approach (0,0).Consider approaching (0,0) along the x-axis. That means fixing y=0 and finding the limit limx→0x2x2+y2. We getlimx→0, y=0x2x2+y2=limx→0x2x2+0limx→0, y=0x2x2+y2=limx→0x2x2limx→0, y=0x2x2+y2=limx→01limx→0, y=0x2x2+y2=1Now, consider approaching (0,0) along the y-axis. This means fixing x=0 and finding the limit limy→0x2x2+y2. We getlimy→0, x=0x2x2+y2=limy→000+y2limy→0, x=0x2x2+y2=limy→00y2limy→0, x=0x2x2+y2=limy→00limy→0, x=0x2x2+y2=0These two different routes to the origin result in two different limits.limx→0, y=0x2x2+y2≠limy→0, x=0x2x2+y2Thus, the original limit does not exist.

Expert answers to "What is the limit of #lim_((x,y)->(0,0))x^2/(x^2+y^2)#?"

Kristian Combs

Answered question

2023-01-04

What is the limit of #lim_((x,y)->(0,0))x^2/(x^2+y^2)#?

Answer & Explanation

sobrevenilac

Beginner2023-01-05Added 8 answers

In order for this limit to exist, the fraction

\frac{x^{2}}{x^{2} + y^{2}}

must approach the same value L, regardless of the path along which we approach

(0, 0)

.Consider approaching (0,0) along the x-axis. That means fixing y=0 and finding the limit

lim_{x \to 0} \frac{x^{2}}{x^{2} + y^{2}} .

We get

lim_{x \to 0, y = 0} \frac{x^{2}}{x^{2} + y^{2}} = lim_{x \to 0} \frac{x^{2}}{x^{2} + 0}

lim_{x \to 0, y = 0} \frac{x^{2}}{x^{2} + y^{2}} = lim_{x \to 0} \frac{x^{2}}{x^{2}}

lim_{x \to 0, y = 0} \frac{x^{2}}{x^{2} + y^{2}} = lim_{x \to 0} 1

lim_{x \to 0, y = 0} \frac{x^{2}}{x^{2} + y^{2}} = 1

Now, consider approaching (0,0) along the y-axis. This means fixing x=0 and finding the limit

lim_{y \to 0} \frac{x^{2}}{x^{2} + y^{2}}

. We get

lim_{y \to 0, x = 0} \frac{x^{2}}{x^{2} + y^{2}} = lim_{y \to 0} \frac{0}{0 + y^{2}}

lim_{y \to 0, x = 0} \frac{x^{2}}{x^{2} + y^{2}} = lim_{y \to 0} \frac{0}{y^{2}}

lim_{y \to 0, x = 0} \frac{x^{2}}{x^{2} + y^{2}} = lim_{y \to 0} 0

lim_{y \to 0, x = 0} \frac{x^{2}}{x^{2} + y^{2}} = 0

These two different routes to the origin result in two different limits.

lim_{x \to 0, y = 0} \frac{x^{2}}{x^{2} + y^{2}} \neq lim_{y \to 0, x = 0} \frac{x^{2}}{x^{2} + y^{2}}

Thus, the original limit does not exist.

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