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.)
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.
$\displaystyle \left| \frac{x+y}{x^2+1} - 0 \right| = \frac{|x+y|}{x^2+1} \leq |x+y| \leq |x|+|y|$
Now if $\displaystyle 0 < \sqrt{x^2+y^2} < \delta \implies |x|,|y|<\delta$
Therefore $\displaystyle |x|+|y| < 2\delta$.
Thus, choose $\displaystyle \delta = \frac{\epsilon}{2}$.