2ху lim (x.y)-(1.0) Which one of the following statements is correct? (a) Consider the limit OThe limit does not exist. OThe limit is infinite. OThe limit exists and is equal to 0). OThe limit exists and is equal to 1. ONone of the previous statements is correct. (b) Consider the limit lim (x)-(0.0) x2 Suppose that you want to test the limit along all lines passing through (0, 0). Which one of the following statements is correct? +y OOne obtains the same limit along all lines, so the limit must exist. OOne obtains the same limit along all lines, but the limit might not exist. OOne obtains two different limits along two different lines, so the limit does not exist. OOne obtains two different limits along two different lines, but the limit might still exist. ONone of the previous statements is correct. (c) Consider the function defined by y f(x, y) = - for (x, y) # (0, 0) and f(0, 0) = +. Which one of the following statements is correct? OThe function is continuous at all points. OThe function is continuous at all points except for (0, 0). OThe function is continuous at all points except for those of the form (x, x) for x € R. OThe function is continuous at all points except for those of the form (x, x) for x €R \ {1, –1}. OThe function is never continuous, because it is not well-defined. ONone of the previous statements is correct. (d) Consider the function n f(x, y) = |x| - y3. Which one of the following statements is correct? OThe function is differentiable at (0,0). OThe function is not differentiable at (0,0), because the partial derivative with respect to x does not exist. OThe function is not differentiable at (0,0), because the partial derivative with respect to y does not exist. OThe function is not differentiable, because the partial derivatives are not defined for some points near (0,0).
2ху lim (x.y)-(1.0) Which one of the following statements is correct? (a) Consider the limit OThe limit does not exist. OThe limit is infinite. OThe limit exists and is equal to 0). OThe limit exists and is equal to 1. ONone of the previous statements is correct. (b) Consider the limit lim (x)-(0.0) x2 Suppose that you want to test the limit along all lines passing through (0, 0). Which one of the following statements is correct? +y OOne obtains the same limit along all lines, so the limit must exist. OOne obtains the same limit along all lines, but the limit might not exist. OOne obtains two different limits along two different lines, so the limit does not exist. OOne obtains two different limits along two different lines, but the limit might still exist. ONone of the previous statements is correct. (c) Consider the function defined by y f(x, y) = - for (x, y) # (0, 0) and f(0, 0) = +. Which one of the following statements is correct? OThe function is continuous at all points. OThe function is continuous at all points except for (0, 0). OThe function is continuous at all points except for those of the form (x, x) for x € R. OThe function is continuous at all points except for those of the form (x, x) for x €R \ {1, –1}. OThe function is never continuous, because it is not well-defined. ONone of the previous statements is correct. (d) Consider the function n f(x, y) = |x| - y3. Which one of the following statements is correct? OThe function is differentiable at (0,0). OThe function is not differentiable at (0,0), because the partial derivative with respect to x does not exist. OThe function is not differentiable at (0,0), because the partial derivative with respect to y does not exist. OThe function is not differentiable, because the partial derivatives are not defined for some points near (0,0).
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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Transcribed Image Text:2ху
lim
(х,у)-(1,0) х2+ y?
(a) Consider the limit
Which one of the following statements is correct?
OThe limit does not exist.
OThe limit is infinite.
OThe limit exists and is equal to ().
OThe limit exists and is equal to 1.
ONone of the previous statements is correct.
(b) Consider the limit
x + y²
lim
Suppose that you want to test the limit along all lines passing through (0, 0). Which one of the following statements is correct?
(х,у)-(0,0) х2 +у
OOne obtains the same limit along all lines, so the limit must exist.
OOne obtains the same limit along all lines, but the limit might not exist.
OOne obtains two different limits along two different lines, so the limit does not exist.
OOne obtains two different limits along two different lines, but the limit might still exist.
ONone of the previous statements is correct.
(c) Consider the function defined by y f(x, y) =
for (x, y) # (0, 0) and f(0, 0) = .
x² + y?
Which one of the following statements is correct?
OThe function is continuous at all points.
OThe function is continuous at all points except for (0, 0).
OThe function is continuous at all points except for those of the form (x, x) for x E R.
OThe function is continuous at all points except for those of the form (x, x) for x E R \ {1, –1}.
OThe function is never continuous, because it is not well-defined.
ONone of the previous statements is correct.
(d) Consider the function n f(x, y) = |x| · y"3. Which one of the following statements is correct?
OThe function is differentiable at (0,0).
OThe function is not differentiable at (0,0), because the partial derivative with respect to x does not exist.
OThe function is not differentiable at (0,0), because the partial derivative with respect to y does not exist.
OThe function is not differentiable, because the partial derivatives are not defined for some points near (0,0).
OThe function is not differentiable, because the partial derivatives are not continuous near (0,0).
ONone of the previous statements is correct.
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