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!


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
6 years, 11 months ago

No vote yet
7 votes

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

(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 - 6 years, 11 months ago

Log in to reply

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 - 6 years, 11 months ago

Log in to reply

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

Ton de Moree - 6 years, 10 months ago

Log in to reply

@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 - 6 years, 10 months ago

Log in to reply

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 - 6 years, 10 months ago

Log in to reply

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

Akbarali Surani - 6 years, 10 months ago

Log in to reply

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

Tamoghna Banerjee - 6 years, 10 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...