l'hopitals rule

• November 20th 2012, 09:58 AM
Greymalkin
l'hopitals rule
Attachment 25825
Can anyone elaborate on that explanation? I don't understand how dividing by zero from a certain side will somehow produce infinity when denominating 1. Whats the difference between 0 and -0? "When approaching 0 from positive numbers with a negative number", how is that supposed to make sense??
• November 20th 2012, 10:11 AM
Greymalkin
Re: l'hopitals rule
also how is it that $csc0=\infty$ yet $csc0=\frac{1}{sin0}$ which equals 1/0 which is undefined, I think these problems are of the same vein.

i've also noticed on some calculators the sin of 0 gives me a really small number instead of 0.
• November 20th 2012, 10:12 AM
skeeter
Re: l'hopitals rule
"explanation" leaves a bit to be desired ...

$\lim_{x \to 0^+} \frac{1}{x} + \frac{1}{\cos{x}-1}$

saying it w/o using the oft misused word "infinity" ...

as $x \to 0^+$ , the first term becomes a very large, unbounded positive value

as $x \to 0^+$ , the second term becomes a very large, unbounded negative value

(large, unbounded positive value) + (large, unbounded negative value) is an indeterminate value ... hence the need to set up the limit ao L'Hopital can be used.
• November 20th 2012, 11:48 AM
Greymalkin
Re: l'hopitals rule
thanks for the explanation but my real query was to why something divided by nothing is equal to an infinite amount of something, or as in the example something infinitely small.
• November 20th 2012, 11:55 AM
Plato
Re: l'hopitals rule
Quote:

Originally Posted by Greymalkin
thanks for the explanation but my real query was to why something divided by nothing is equal to an infinite amount of something, or as in the example something infinitely small.

Would you agree that $\frac{1}{n}$ gets smaller as we make $n$ larger?

If so, consider $\frac{1}{\frac{1}{n}}=n$ which gets larger and larger as $\frac{1}{n}$ gets smaller and smaller.
• November 20th 2012, 11:56 AM
skeeter
Re: l'hopitals rule
Quote:

Originally Posted by Greymalkin
thanks for the explanation but my real query was to why something divided by nothing is equal to an infinite amount of something, or as in the example something infinitely small.

this is the concept that you have the wrong idea about ... you're not dividing by "nothing", you're dividing a fixed value by an infinitesimally small value.

understand that you're finding the limit as x approaches zero , not at x = 0.
• November 20th 2012, 07:19 PM
divans
Re: l'hopitals rule
You could also think about the graph of 1/x and consider what happens to the y values as x gets closer to zero.