Help with High-Power Limit Problem

  • Context: MHB 
  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Limit
Click For Summary
SUMMARY

The discussion focuses on evaluating the limit as \( x \) approaches infinity for the expression \(\lim_{x\to \infty }\frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}\). A key insight shared is to simplify the expression by dividing each term by \( x \), leading to the transformation \(\frac{(2-x^{-1})^{10}(1-2x^{-1})^{30}}{(1+x^{-1})^{20}(2+3x^{-1})^{20}}\). This method highlights that the highest power, \( x^{30} \), dominates the behavior of the limit as \( x \) approaches infinity.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with polynomial expressions and their degrees
  • Knowledge of asymptotic behavior of functions
  • Ability to manipulate algebraic expressions involving powers
NEXT STEPS
  • Study techniques for evaluating limits at infinity
  • Learn about polynomial long division and its applications
  • Explore L'Hôpital's Rule for indeterminate forms
  • Investigate the concept of dominant terms in rational functions
USEFUL FOR

Students and educators in calculus, mathematicians focusing on limits, and anyone looking to deepen their understanding of asymptotic analysis in polynomial functions.

Yankel
Messages
390
Reaction score
0
hello,

I need some help with this limit...

\lim_{x\to \infty }\frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}

thanks...
 
Physics news on Phys.org
What ideas have you had so far?
 
The only thing I could think of was to somehow push x^30 inside the powers, but I don't know how...

30 is the strongest power when it comes to infinity, so there must be something with 30.
 
Yankel said:
hello,

I need some help with this limit...

\lim_{x\to \infty }\frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}

thanks...
for \( x\ne 0 \) divide each of the 40 brackets at top and bottom by \(x\):

\[ \frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}= \frac{(2-x^{-1})^{10}(1-2x^{-1})^{30}}{(1+x^{-1})^{20}(2+3x^{-1})^{20}}\]

CB
 

Similar threads

Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
3K
High School Wondering if a limit exists or not
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Calculate limits as distributions
  • · Replies 4 ·
Replies
4
Views
2K
MHB 1.6.1 AP Calculus Exam Limits with L'H
  • · Replies 3 ·
Replies
3
Views
2K
MHB What is the Epsilon-Delta Method for Proving Limits?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
2K
High School Rationale Behind t-Substitution for Evaluating Limits?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Using limits to solve an arithmetic problem?
  • · Replies 4 ·
Replies
4
Views
2K
High School Naturals or Reals When Taking Limits to Obtain the Value of Euler's Number e?
  • · Replies 11 ·
Replies
11
Views
3K
MHB Limits with the natural logarithm
  • · Replies 3 ·
Replies
3
Views
2K