I am still confused about limit?

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

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

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

Prove your answer!

PS:

I don't know how can I prove it. But I know that limxcf(x)=L\lim_{x \to c} f(x) = L exist if and only if limxc+f(x)=L\lim_{x \to c^+} f(x) = L and limxcf(x)=L.\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
5 years, 11 months ago

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7 votes

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

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

At 0, ff 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)=x2x(x<0) f(x) = \frac {x^2}{x} (x < 0) f(x)=x+3(x=0)f(x) = x + 3 (x = 0) f(x)=x2(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: https://i.imgur.com/2Gh4qJF.png

The enclosed red circle indicates f(0)f(0), i.e., what the function is when evaluated at 00. 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 - 5 years, 11 months ago

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Another example (if you want) of a function like above is: f(x)={2x=01x0 f(x) = \left\{ \begin{array}{l l} 2 & \quad x = 0\\ 1 & \quad x \neq 0 \end{array} \right.

Even though limx0f(x)=1 \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 limxcf(x)=L \lim_{x \to c} f(x) = L mean that f(c)=L f(c) = L for continuous functions?

Kim Laberinto - 5 years, 11 months 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 - 5 years, 11 months ago

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

Michael Tang - 5 years, 11 months 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 - 5 years, 11 months 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 - 5 years, 11 months ago

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

Tamoghna Banerjee - 5 years, 11 months ago

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