Answered: 59. Does knowing that x*y? tan xy 3 ху…

59. Does knowing that x*y? tan xy 3 ху tell you anything about tan¬ xy lim (х, у) —- (0,0) ху Give reasons for your answer.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 19E

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Question

59. Does knowing that
x*y?
tan xy
3
ху
tell you anything about
tan¬ xy
lim
(х, у) —- (0,0)
ху
Give reasons for your answer.

Expert Solution

Step 1

The limit of a function of two variables $f (x, y)$ is L when $(x, y)$ approaches to $(x_{0}, y_{0})$ and we denote it

by $\lim_{(x, y) \to (x_{0}, y_{0})} f (x, y) = L$ . It is defined as for any $ε > 0$ , there exists $δ > 0$ such that for all $(x, y) \neq (x_{0}, y_{0})$

if $(x, y)$ is in the open disc centred at $(x_{0}, y_{0})$ with radius $δ$ , then $|f (x, y) - L| < ε$ .

We have to find the limit of the function $\frac{\tan^{- 1} (x y)}{x y}$ at the point $(0, 0)$ when it is given that the inequality

$1 - \frac{x^{2} y^{2}}{3} < \frac{\tan^{- 1} (x y)}{x y} < 1$ holds.

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