Determine whether the statement is true or false. If it is false, explain why or give...

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If f(c) = L,then lim x→c f(x) = L.

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 f(−1) = 2 while the limit of f as x approaches −1 is equal to −2.

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 f(2) = 0 while the limit of f as x approaches 2 is equal to −2.

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

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

The statement is true.

Solutions

Expert Solution

If f(c) =L then it is (False)

We can see by an example -

Here

So these are not equal.

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 f(−1) = 2 while the limit of f as x approaches −1 is equal to −2. (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 f(2) = 0 while the limit of f as x approaches 2 is equal to −2. (True)

And

If f(c) = L, then the limit of f as x approaches c is equal to L/c. (False) There is no relation between these.

False. If f(c) = L, then the limit of f as x approaches c is equal to cL. (False) There is no relation between these.

milcah answered 1 year ago

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