Using the delta-epsilon definition of the limit -- solve this problem, please!

taureau20 · #1 $Old$ September 26th, 2008, 08:36 AM

Show that there exists a d>0, such that f has a limit at (0,0). Find the value of d.
f(x,y) = (x + y)/(x2 + 1) ; e = 0.01
Answer: d = 0.005
(Problem number 63, exercise, 12.2 from Calculus (9th edition by Thomas and Finney.)

ThePerfectHacker · #2 $Old$ September 26th, 2008, 11:19 AM

Quote:

Originally Posted by taureau20 $View Post$

Show that there exists a d>0, such that f has a limit at (0,0). Find the value of d.
f(x,y) = (x + y)/(x2 + 1) ; e = 0.01
Answer: d = 0.005
(Problem number 63, exercise, 12.2 from Calculus (9th edition by Thomas and Finney.)

We will show the limit is zero.

$\left| \frac{x+y}{x^2+1} - 0 \right| = \frac{|x+y|}{x^2+1} \leq |x+y| \leq |x|+|y|$

Now if $0 < \sqrt{x^2+y^2} < \delta \implies |x|,|y|<\delta$

Therefore $|x|+|y| < 2\delta$ .

Thus, choose $\delta = \frac{\epsilon}{2}$ .