Limit Problem: Itermediate Value Theorem

In summary, the conversation discusses using the Intermediate Value Theorem and the definition of limit at infinity to show that for a continuous function with a limit of -1 at negative infinity and a limit of 10 at positive infinity, there must be at least one point where the function equals 0. The conversation also mentions applying this concept to prove that every odd degree polynomial has a root.
  • #1
MrBailey
19
0
Hello out there. I hope everyone is doing well.
I could use a little guidance on this:

suppose [tex]f[/tex] is continuous for all [tex]x[/tex], and

[tex]\lim_{x\rightarrow -\infty}f(x) = -1[/tex] and [tex]\lim_{x\rightarrow +\infty}f(x) = 10[/tex]

Show that [tex]f(x) = 0[/tex] for at least one [tex]x[/tex]

I know I need to use the Intermediate Value Theorem and the definition of the limit...but I'm not really sure how to apply them.

Thanks,
Bailey
 
Physics news on Phys.org
  • #2
The definition of a limit at infinity will give you some finite value of x for which f(x) is within some neighborhood of 10 (and, seperately, -1). Then use IVT.
 
  • #3
thanks!

I see it now.

Bailey
 
  • #4
try to prove then that every odd degree polynomial has a root.
 
  • #6
it was homework problem in frosh calc that i did not get at the time.
 

FAQ: Limit Problem: Itermediate Value Theorem

What is the Intermediate Value Theorem (IVT)?

The Intermediate Value Theorem is a mathematical theorem that states that if a function is continuous on a closed interval [a, b], then for any value c between f(a) and f(b), there exists at least one value x in the interval [a, b] such that f(x) = c.

How is the Intermediate Value Theorem used to solve limit problems?

The Intermediate Value Theorem can be used to prove the existence of a limit for a function at a specific point. If we can show that a function is continuous at that point, then by the IVT, we can conclude that the limit exists.

Can the Intermediate Value Theorem be applied to all functions?

No, the Intermediate Value Theorem only applies to continuous functions. This means that the function must have no breaks or jumps in its graph and must have a continuous line from one point to another.

How can we determine if a function is continuous at a specific point?

A function is continuous at a point if the limit from the left side of the point is equal to the limit from the right side of the point, and both limits exist. This means that the function is smooth and has no gaps or holes at that point.

What is the difference between the Intermediate Value Theorem and the Mean Value Theorem?

The Intermediate Value Theorem states that for a continuous function on a closed interval, there exists at least one point where the function takes on a specific value. The Mean Value Theorem, on the other hand, states that for a differentiable function on an open interval, there exists at least one point where the slope of the tangent line is equal to the average rate of change of the function over the interval.

Similar threads

Evaluate the limit of this harmonic series as n tends to infinity
Replies
17
Views
2K
Limit Definition of Derivative as n Approaches Infinity
Replies
20
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
Replies
6
Views
1K
Mean Value Theorem: Proof & Claim
Replies
11
Views
2K
Extreme value theorem, proof question
Replies
12
Views
2K
Problems solving a limit which results in an indeterminate form
Replies
4
Views
1K
Determine the limit of 2^x/x^2 as x approaches infinity....
Replies
26
Views
2K
Show that the limit (1+z/n)^n=e^z holds
Replies
6
Views
1K
Jacobian: how to change limits of integration?
Replies
16
Views
2K
What is the Definition of Derivative and How is it Proved?
Replies
7
Views
2K

Hot Threads

Recent Insights

Back
Top