Thursday, 31 October 2019

Limit limxrightarrowinfty,yrightarrowinftyleft(fracxyx2+y2right)x2

Given the followning limit:
lim



To find limit I have made following steps:




  1. Let x = y ,then limit equals 0

  2. Let x > y ,then consider the limit:




\lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{x^2}{x^2 + y^2}\right)^{x^2} = \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{1}{1 + \frac{y^2}{x^2}}\right)^{x^2} = 0
with respect to 0 < y^2/x^2 < const




  1. Let y > x ,then consider the limit:



\lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{x^2}{x^2 + y^2}\right)^{y^2} = \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{1}{\frac{x^2}{y^2} + 1}\right)^{x^2} = 0
with respect to 0 < x^2/y^2 < const




What could you say about my solution?

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