Find a value of constant k so a limit to infinity exists

Click For Summary
SUMMARY

The discussion focuses on determining the constant value of "k" for the limit of the expression (x^3 - 6) / (x^k + 3) as x approaches infinity. It is established that k = 3 allows the limit to exist, resulting in a value of 1. For k > 3, the limit approaches 0, indicating existence, while for k < 3, the limit approaches infinity, indicating non-existence. The key takeaway is that the growth rates of the numerator and denominator must be similar for the limit to exist.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with polynomial functions
  • Knowledge of asymptotic behavior of functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the concept of limits at infinity in calculus
  • Learn about polynomial growth rates and their implications
  • Explore the application of L'Hôpital's Rule for indeterminate forms
  • Investigate the behavior of rational functions as x approaches infinity
USEFUL FOR

Students studying calculus, particularly those focusing on limits and asymptotic analysis, as well as educators seeking to clarify concepts related to polynomial limits.

madgab89
Messages
22
Reaction score
0
Find a value of constant "k" so a limit to infinity exists

Homework Statement



Find a value of the constant k such that the limit exists.

Homework Equations



lim x to infinity of

x^3-6/x^k+3

The Attempt at a Solution


I started by setting the equation equal to infinity and attempted to rearrange it but got pretty much nowhere. I also broke it up using limit rules and also ended up nowhere.
 
Physics news on Phys.org


Can you guess what a k might be? For the limit to exist the numerator and denominator have to grow at similar rates as x->infinity.
 


So would k be 3? I'm not sure I'm understanding..
 


Yes, k=3 is one value that works. What's the limit in that case? Can you show if k>3 the limit is 0 (so it exists) and if k<3 the limit is infinity (so it doesn't exist)? You were just asked to find 'a value'. There are lots of choices.
 


oh my goodness. i just understood it now..thank you *bangs head on desk* :)
 


so to show that say for k=4 the limit is 0, would I just divided each term top and bottom by x^4?
 


and for k=3 the value of the limit would be 1, correct?
 


madgab89 said:
and for k=3 the value of the limit would be 1, correct?

Right on both counts.
 

Similar threads

Can the limit of a quotient of trig functions approach a specific value?
  • · Replies 2 ·
Replies
2
Views
1K
Help Taking the Limit as K goes to infinity
  • · Replies 14 ·
Replies
14
Views
3K
Limit of Taylor Polynomial for Tn(x) as n Approaches Infinity
  • · Replies 2 ·
Replies
2
Views
1K
Multivariable Limit Problem: Find Values of k That Make Limit Exist
  • · Replies 9 ·
Replies
9
Views
2K
Limit of x^α.sin²(x)/(x+1) as x approaches infinity is?
  • · Replies 13 ·
Replies
13
Views
3K
Limit of (-x^(k+1))/e^x: Solving Step by Step with l'Hopital's Rule
  • · Replies 5 ·
Replies
5
Views
2K
Sinusoidal sequences with random phases
  • · Replies 1 ·
Replies
1
Views
1K
Prove that the limit of a sequence exists
  • · Replies 17 ·
Replies
17
Views
3K
Find the limit of the function
  • · Replies 8 ·
Replies
8
Views
2K
Prove the sequence has a limit value and find it's limit value
  • · Replies 5 ·
Replies
5
Views
2K