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August 30th, 2016, 03:37 PM   #1
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Is this true or false? Please explain also so I can learn.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If lim x→c f(x) = L,then f(c) = L.

a) False. If the limit of f as x approaches c is equal to L, then f(c) = cL.

b) False. Define f to be the piece-wise function where f(x) = x + 3 when x ≠ −1 and f(x) = 2 when x = −1. Then we have that the limit of f as x approaches −1 is equal to −2, while f(−1) = 2.

c) False. Define f to be the piece-wise function where f(x) = x − 4 when x ≠ 2 and f(x) = 0 when x = 2. Then we have that the limit of f as x approaches 2 is equal to −2, while f(2) = 0.

D) False. If the limit of f as x approaches c is equal to L, then f(c) = L/c.
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August 30th, 2016, 05:04 PM   #2
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All choices are false ... what does that tell you?

Correct response is depicted in the attached graph.
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File Type: jpg piecewise.jpg (7.7 KB, 11 views)
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August 30th, 2016, 10:31 PM   #3
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What have you learned about limits? It makes a difference as to how it might be best explained.

Clearly, the initial statement ($\lim \limits_{x \to c} f(x) = L \implies f(c)=L$) is false because all the options you give state that it is false. The question is, can you say why?

I don't understand why anyone would pick answers a) and d), they don't sound anything like any part of the theory of limits as I understand it. One of b) and c) is correct, but if you have any understanding of limits it shouldn't be too difficult to decide which is correct.

What do you think the answer should be, and why?
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September 1st, 2016, 02:43 PM   #4
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If the function f is not continuous at x=c then
limf(x) is not equal to f(c).
x->c
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