Evaluate the following: To evaluate the limit of the expression lim(x,y) →(0,0) (xy + 1)/(x^2 + y^2 + 1), we can approach the point (0,0) along different paths and see if the limit is the same regardless of the path taken. Let's consider two paths: the x-axis and the y-axis. 1. Approach along the x-axis (y = 0): lim(x,0)→(0,0) (x*0 + 1) / (x^2 + 0^2 + 1) = lim(x,0)→(0,0) 1/(x^2 + 1) = 1/(0^2 + 1) = 1. 2. Approach along the y-axis (x = 0): lim(0,y) →(0,0) (0*y + 1) / (0^2 + y^2 + 1) = lim(0,y) →(0,0) 1 / (y^2 + 1) = 1/(0^2 + 1) = 1. Since the limit is the same along both paths (1 in this case), we can conclude that the limit of the given expression as (x, y) approaches (0, 0) is 1. The approach is correct The approach is incorrect The approach is almost acceptable
Evaluate the following: To evaluate the limit of the expression lim(x,y) →(0,0) (xy + 1)/(x^2 + y^2 + 1), we can approach the point (0,0) along different paths and see if the limit is the same regardless of the path taken. Let's consider two paths: the x-axis and the y-axis. 1. Approach along the x-axis (y = 0): lim(x,0)→(0,0) (x*0 + 1) / (x^2 + 0^2 + 1) = lim(x,0)→(0,0) 1/(x^2 + 1) = 1/(0^2 + 1) = 1. 2. Approach along the y-axis (x = 0): lim(0,y) →(0,0) (0*y + 1) / (0^2 + y^2 + 1) = lim(0,y) →(0,0) 1 / (y^2 + 1) = 1/(0^2 + 1) = 1. Since the limit is the same along both paths (1 in this case), we can conclude that the limit of the given expression as (x, y) approaches (0, 0) is 1. The approach is correct The approach is incorrect The approach is almost acceptable
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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