How to Prove the Limit of a Function at a Point with Multiple Paths?

[mathnerd15]

Apostol Limit Problem?

Homework Statement

I can't afford the Apostol calculus vol. 2 there's a printing mistake in my copy of Apostol and I'm not sure how to prove this, p.251

let f(x,y)={xsin(1/y) if y doesn't equal 0
and f=0 if y=0
prove that the iterated limits are not equal and that the f(x,y)->0 as (x,y)->(0,0)

Homework Equations

The Attempt at a Solution

how exactly do you prove the limit for (x,y)->0 from all possible paths, parabolic paths, x=y, polar paths?

 

[pasmith]

Homework Statement

I can't afford the Apostol calculus vol. 2 there's a printing mistake in my copy of Apostol and I'm not sure how to prove this, p.251

let f(x,y)={xsin(1/y) if y doesn't equal 0
and f=0 if y=0
prove that the iterated limits are not equal and that the f(x,y)->0 as (x,y)->(0,0)

If this is exercise 5 of section 8.5, then what you have corresponds to my text (save that it's on page 252.)

Homework Equations

The Attempt at a Solution

how exactly do you prove the limit for (x,y)->0 from all possible paths, parabolic paths, x=y, polar paths?

Use
<br /> |x \sin y^{-1}| \leq |x| \leq \sqrt{x^2 + y^2}<br />