Chapter 12: Q. 12 (page 989)

Evaluate the following limits, or explain why the limit does not exist.
$\lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}}$

Short Answer

Expert verified

$\lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = 1$

Step by step solution

Step 1. Given information is:

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

Step 2. Evaluating Limits

$As the given limit \lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}} has to be evaluated alon the line y = 0, so \lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \lim_{y \to 0} \frac{x^{2} + 0^{2}}{x^{2} - 0^{2}} \lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \lim_{x \to 0} (1) \lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = 1 Therefore, \lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = 1$

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Evaluate the following limits, or explain why the limit does not exist.
$\lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}}$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating Limits

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Evaluate the following limits, or explain why the limit does not exist.lim(x,y)→(0,0)y=0x2+y2x2-y2

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Evaluating Limits

One App. One Place for Learning.

Most popular questions from this chapter

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Evaluate the following limits, or explain why the limit does not exist.
$\lim_{\frac{(x, y) \to (0, 0)}{y = 0}} \frac{x^{2} + y^{2}}{x^{2} - y^{2}}$