Solving Limits and Continuity Problems with Examples and Proofs

In summary, the conversation covers various topics related to continuity in functions. The first question asks for help in finding a constant a that will make a function continuous on the entire real line. The second question involves the Intermediate Value Theorem and finding a specific value of c that is guaranteed by the theorem. The third question discusses the statement "if f(x)=g(x) for x≠c and f(c)≠g(c) then either f or g is not continuous at c" and whether it is true or false. Lastly, the conversation touches on the proof of continuity using the definition of continuity and the question of whether a given function is continuous at any real number.
  • #1
vbplaya
11
0
I'm having a little trouble trying to figure out these problems. Any help would be appreciated.

g(x) = (x^2 - a^2)/(x-a) when x≠a but 8 when x=a... how do i find the constant a so that the function will be continuous on the entire real line?


f(x)= x^3 - x^2 + x - 2 on closed interval [0,3] f(c)=4. How do I find the value of c that is guaranteed by the Intermediate Value Theorem?
---I've proven via IVT that there exists a 0 in [0,3] but I do not know how to find the c value.


if f(x)=g(x) for x≠c and f(c)≠g(c) then either f or g is not continuous at c. True or False.
--- I haven't a clue. I can't even think of an example where f(x)=g(x) but f(c)≠g(c).


this last one I just want to make sure I'm doing it right.
Show that the Dirichlet function f(x)= 0 if x is rational and 1 if x is irrational
is not continuous at any real number.

if I just write D(x) = lim m→∞ lim n→∞ cos^2n (m! pi x) is that showing that the function is not continuous at any real number?

Again, I'd appreciate any help or pointers in the right direction. Thanks.
 
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  • #2
cant u simplify the first one such that you get g(x) = x+a and a cna be any real number?

ALso for the continuity proof
use the DEFINITION OF CONTINUITY that is
[tex] \lim_{x \rightarrow c} f(x) = f(c) [/tex] if [tex] \forall \epsilon >0, \exists \delta>0 [/tex] such that if [tex] 0<|x-c|< \delta [/tex] then [tex] |f(x)-f(c)| <\epsilon [/tex]

you have to use the definition to prove continuity. Who says that that formula u wrote is continuous or not?
 

1. What are limits and continuity in mathematics?

Limits and continuity are fundamental concepts in mathematics that deal with the behavior of a function as the input values approach a certain point. A limit is the value that a function approaches as the input values get closer and closer to a specific value. Continuity refers to the property of a function where there are no abrupt changes or breaks in the graph.

2. How do you solve a limit problem?

To solve a limit problem, you can use different techniques such as direct substitution, factoring, and rationalization. Direct substitution involves plugging in the value that the function is approaching. Factoring and rationalization are methods used to simplify the function and eliminate any potential discontinuities. Additionally, you can use graphs and tables to visualize and approximate the limit value.

3. What are the three types of limits?

The three types of limits are one-sided limits, two-sided limits, and infinite limits. One-sided limits are when the input values approach a specific value from either the left or the right. Two-sided limits are when the input values approach a specific value from both sides. Infinite limits occur when the output of a function approaches positive or negative infinity as the input values approach a certain value.

4. How do you determine if a function is continuous?

A function is continuous if it is defined at a specific point, the limit of the function at that point exists, and the value of the function at that point is equal to the limit. In other words, the graph of a continuous function has no breaks or holes, and you can draw it without lifting your pencil from the paper.

5. Why is it important to prove continuity and limits?

Proving continuity and limits is important because it allows us to understand the behavior of a function and make accurate predictions about its values. It also helps us determine the existence of discontinuities, which may indicate errors in our calculations or assumptions. Furthermore, continuity and limits are essential concepts in calculus and many other branches of mathematics, making them crucial for solving more complex problems.

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