Chapter 12: Q. 15 (page 931)

Provide a definition for $\lim_{(x, y) \to (a, b)} f (x, y) = \infty$ . Model your definition on Definitions 1.9 and 12.15.

Short Answer

Expert verified

For all $M > 0$ ,there exist a $δ > 0$ such that if $0 < \sqrt{{(x - a)}^{2} + {(y - b)}^{2}} < δ$ , then $f (x) \in (M, \infty)$ .

Step by step solution

Defining the limit

The goal is to provide a definition for $\lim_{(x, y) \to (a, b)} f (x, y) = \infty$ .

The infinite limit at a point, represented as $\lim_{(x) \to (a)} f (x) = \infty$ , denotes the existence of an infinite limit for every $M > 0$ , such that if $x \in (a - δ, a) \cup (a, a + δ)$ , then $f (x) \in (M, \infty)$

The limit of a two-variable function fis L, expressed as $\lim_{(x) \to (a)} f (x) = L$ if, for every $ε > 0$ , there is $δ > 0$ such that $|f (x) - L| < ε$ whenever role="math" localid="1653486855833" $0 < ||x - a|| < δ$ .

Evaluating the limit

To define the infinite limit of a function in two variables, combine the two definitions.

The infinite limit of a function in two variables at a point, represented as $\lim_{(x, y) \to (a, b)} f (x, y) = \infty$ , indicates that for any $M > 0$ , there exists $δ > 0$ such that if is $0 < \sqrt{{(x - a)}^{2} + {(y - b)}^{2}} < δ$ , then $f (x) \in (M, \infty)$ .

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Provide a definition for $\lim_{(x, y) \to (a, b)} f (x, y) = \infty$ . Model your definition on Definitions 1.9 and 12.15.

Short Answer

Step by step solution

Defining the limit

Evaluating the limit

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Provide a definition for lim(x,y)→(a,b)f(x,y)=∞. Model your definition on Definitions 1.9 and 12.15.

Short Answer

Step by step solution

Defining the limit

Evaluating the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Provide a definition for $\lim_{(x, y) \to (a, b)} f (x, y) = \infty$ . Model your definition on Definitions 1.9 and 12.15.