1. Investigate the limiting behaviour ofx2lim(x,y)¬(0,0) x² + 3xy + y²

The limit of the function is, limx,y→0,0x2x2+3xy+y2 As x,y→0,0 we see that the given limit is of the…

Answered: Investigate the limit

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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Question

1. Investigate the limiting behaviour of
x2
lim
(x,y)¬(0,0) x² + 3xy + y²

Expert Solution

Step 1

The limit of the function is,

$\lim_{(x, y) \to (0, 0)} \frac{x^{2}}{x^{2} + 3 x y + y^{2}}$

As $(x, y) \to (0, 0)$ we see that the given limit is of the form $\frac{a}{b}$ . So we can apply the L'hospital rule .

$\begin{array}{rcl} \lim_{(x, y) \to (0, 0)} \frac{x^{2}}{x^{2} + 3 x y + y^{2}} & = & \lim_{(x, y) \to (0, 0)} \frac{2 x}{2 x + 3 (y + x) + 2 y} \\ = & \lim_{(x, y) \to (0, 0)} \frac{2 x}{5 (y + x)} \\ = & \lim_{(x, y) \to (0, 0)} \frac{2}{5 (1 + 1)} \\ = & \frac{1}{10} \end{array}$

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