Dominant Terms in Calculus Limits

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SUMMARY

The dominant term in the numerator of the expression lim [(x^7)-9(e^x)] / [sqrt(10x-1)+8*ln(x)] as x approaches infinity is -9(e^x), while the dominant term in the denominator is sqrt(10x-1). Consequently, the limit evaluates to -∞. This conclusion contradicts initial assumptions regarding the dominance of x^7 in the numerator and sqrt(10x-1) in the denominator, highlighting the importance of recognizing exponential growth over polynomial and logarithmic terms.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with exponential functions and their growth rates
  • Knowledge of polynomial and logarithmic functions
  • Basic skills in evaluating limits using dominance
NEXT STEPS
  • Study the concept of dominance in limits, focusing on exponential vs. polynomial growth
  • Learn techniques for evaluating limits involving indeterminate forms
  • Explore the properties of logarithmic functions in limit calculations
  • Review examples of limits approaching infinity with varying dominant terms
USEFUL FOR

Students studying calculus, particularly those focusing on limits and dominance in mathematical expressions, as well as educators seeking to clarify these concepts in instructional settings.

spingo
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Hello, I am having issues finding the dominant terms in the following expression:

lim [(x^7)-9(e^x)] / [sqrt(10x-1)+8*ln(x)]
x->infinity

Prompt: Find the limit and the dominant term in the numerator and denominator.
 
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Link to a decent explanation of dominance ...

Dominance
 
So the dominant term in the numerator is should be (x^7)? According to my assignment this is incorrect but I'm not sure how. Additionally, by the logic used in the document sqrt(10x-1) should be dominant in the denominator.

The limit should be infinity then?

My assignment claims all of these answers are incorrect.
 
spingo said:
So the dominant term in the numerator is should be (x^7)? According to my assignment this is incorrect but I'm not sure how. Additionally, by the logic used in the document sqrt(10x-1) should be dominant in the denominator.

The limit should be infinity then?

My assignment claims all of these answers are incorrect.

dominant term in the numerator is the exponential, as stated clearly in the link.

yes, dominant term in the denominator is the square root function

the limit is $-\infty$
 

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