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Chapter 12: Q. 33 (page 931)

Evaluate the limits in Exercises 33鈥�40 if they exist.

lim(x,y)鈫�(-2,蟺)x2y3siny

Short Answer

Expert verified

The limit is0.

Step by step solution

01

Given Information

Consider the phrase lim(x,y)鈫�(-2,蟺)x2y3siny

02

Defining the limit 

The goal is to assess lim(x,y)鈫�(-2,蟺)x2y3sinyif it exists.

Consider the following assertion: Consider a two-variable function f(x,y)that is continuous at all points on R2. The limit of the function f(x,y)as (x,y)鈫�(x0,y0)thus defined as lim(x,y)鈫�(x0,y0)f(x,y)=f(x0,y0)

03

Evaluating the limit

Because x2y3is a two-variable polynomial function, it is continuous for every point on R2, and the transcendental number sinyis also continuous for every point on R2.

As a result of the statement,

lim(x,y)鈫�(-2,蟺)x2y3siny=(-2)2(蟺)3sin蟺=0

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