Finite limit problem

  • Thread starter lukatwo
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  • #1
lukatwo
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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.
 

Answers and Replies

  • #2
36,710
8,711

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?
$$\lim_{x \to 0} \frac{cos(x)}{x^0} $$
, 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.

Homework Statement





Homework Equations





The Attempt at a Solution

 
  • #3
lukatwo
24
0
It was a typo. Fixed, and yes that is the limit i was referring to!
 
  • #4
36,710
8,711
As long as x ≠ 0, x0 = 1, right? So, then, what is ##\lim_{x \to 0} x^0##?
 
  • #5
lukatwo
24
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?
 
  • #6
36,710
8,711
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).
 
  • #7
lukatwo
24
0
I understand now! Thank you very much for your help!
 

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