Finite Limit Problem: Understanding cosx/x^0 = 1

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SUMMARY

The limit of the expression lim x->0 (cos(x)/x^0) equals 1, as confirmed by both textbook references and WolframAlpha. The confusion arises from the interpretation of x^0, which equals 1 for all x ≠ 0. The discussion clarifies that as x approaches 0, the value of x^0 remains 1, leading to the conclusion that lim x->0 (cos(x)/x^0) = 1. This understanding is crucial for resolving the misconception that the limit might be undefined or infinite.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with the concept of indeterminate forms
  • Basic knowledge of trigonometric functions, specifically cosine
  • Awareness of L'Hôpital's Rule and its application
NEXT STEPS
  • Study the properties of limits, particularly lim x->0 for various functions
  • Explore the concept of indeterminate forms and how to resolve them
  • Learn about the behavior of exponential functions, especially x^0
  • Review the application of L'Hôpital's Rule in different limit scenarios
USEFUL FOR

Students studying calculus, educators teaching limits, and anyone seeking to clarify the behavior of functions as they approach zero.

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Homework Statement



I've read in my textbook, and confirmed via WolframAlpha that lim x->0 (cosx/x^0) =1 , and need an explanation for it. I thought it should be ∞ or something undefined, since 0^0 is undefined.

Homework Equations


The Attempt at a Solution



I tried to use L'Hospitale on the expression, but that led to nowhere. There's no intermediate step between the expression and solution in both textbook, and Wolfram.
 
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lukatwo said:

Homework Statement



I've read in my textbook, and confirmed via WolframAlpha that lim n->0 (cosx/x^0) =1
Typo?
There is no n in your limit expression.

Is this the limit?
$$\lim_{x \to 0} \frac{cos(x)}{x^0} $$
lukatwo said:
, and need an explanation for it. I thought it should be ∞.

Homework Equations





The Attempt at a Solution



I tried to use L'Hospitale on the expression, but that led to nowhere. There's no intermediate step between the expression and solution in both textbook, and Wolfram.
 
It was a typo. Fixed, and yes that is the limit i was referring to!
 
As long as x ≠ 0, x0 = 1, right? So, then, what is ##\lim_{x \to 0} x^0##?
 
From all this my guess will be 1, but still not clear why. Is it because lim x->0 means that x !=0 but rather close to 0? Meaning some infinitesimal number ^0=1? Am i getting that correctly?
 
lukatwo said:
From all this my guess will be 1, but still not clear why. Is it because lim x->0 means that x !=0 but rather close to 0? Meaning some infinitesimal number ^0=1? Am i getting that correctly?
Yes to both questions. The graph of y = x0 is a horizontal line with a hole at (0, 1).
 
I understand now! Thank you very much for your help!
 

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