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I am still confused about limit?

If given \(f(c) = L,\)

(a) should \(\lim_{x \to c} f(x)\) exist?

(b) If the limit is exist, should \(\lim_{x \to c} f(x) = L?\)

Prove your answer!

PS:

I don't know how can I prove it. But I know that \(\lim_{x \to c} f(x) = L\) exist if and only if \(\lim_{x \to c^+} f(x) = L\) and \(\lim_{x \to c^-} f(x) = L.\) Hence (b) is not true. But then my teacher said that was an incomplete solution. Can anyone help?

Note by Nabila Nida Rafida
4 years, 1 month ago

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(a) The function being defined at \(c\) does not mean that the limit is defined at \(c\). For example, given the piecewise function

\[f(x) = \frac{1}{x} (x < 0)\] \[ f(x) = 5 - x(x = 0)\] \[f(x) = sin(\frac{1}{x}) (x > 0)\]

At 0, \(f\) is defined (it equals 5), but the limit does not exist at 0 because the left and right sided limits are unequal (and don't exist).

(b) No. Again, with a piecewise function

\[ f(x) = \frac {x^2}{x} (x < 0)\] \[f(x) = x + 3 (x = 0)\] \[ f(x) = x^2 (x > 0)\]

The limit of this function at 0 is in fact 0 since the left sided limit is 0 as well as the right sided limit. However, when this function is defined at 0, it is not 0, in fact, it is 3.

Accompanying graph: http://i.imgur.com/2Gh4qJF.png

The enclosed red circle indicates \(f(0)\), i.e., what the function is when evaluated at \(0\). However, the empty red circle indicates the limit at 0, i.e., the mutual value that the functions to the left and right side of it are approaching.

Michael Tong - 4 years, 1 month ago

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Wow, thanks for your quick answer Michael T. I answer the same, saying both questions with no. But I wonder, are there any proof without giving counterexample?

Nabila Nida Rafida - 4 years, 1 month ago

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Another example (if you want) of a function like above is: \[ f(x) = \left\{ \begin{array}{l l} 2 & \quad x = 0\\ 1 & \quad x \neq 0 \end{array} \right.\]

Even though \( \lim_{x \to 0} f(x) = 1 \), f(0) is 2.

Hmmm I pose a question though. In the original question, the function didn't have to be continuous, but what if it did? If the function is continuous, would the statements above always be true? Would the \( \lim_{x \to c} f(x) = L \) mean that \( f(c) = L \) for continuous functions?

Kim Laberinto - 4 years, 1 month ago

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I believe that's actually one of the definitions of a continuous function. I'm a bit rusty on that though.

Ton De Moree - 4 years, 1 month ago

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@Ton De Moree Yep, a function \(f(x)\) is continuous iff \(\lim_{x \to c} f(x) = f(c)\) for all \(c\) (in the domain of \(f.\))

Michael Tang - 4 years, 1 month ago

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To know the basics of limits, i suggest u to join a free online course of calculus one on coursera.org

Akbarali Surani - 4 years, 1 month ago

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Then limit must be continuous for limx→cf(x)=L.

Tamoghna Banerjee - 4 years, 1 month ago

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