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

[lukatwo]

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.

 

[Mark44]

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.

 

[lukatwo]

It was a typo. Fixed, and yes that is the limit i was referring to!

 

[Mark44]

As long as x ≠ 0, x⁰ = 1, right? So, then, what is $\lim_{x \to 0} x^0$?

 

[lukatwo]

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?

 

[Mark44]

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).

 

[lukatwo]

I understand now! Thank you very much for your help!