Question

limit as x approach 0 of sinx divide by x

Answer 1

the answer would be 1

Answer 2

use l'hopital's rule

Answer 3

\[\lim_{x \rightarrow 0}\frac{sinx}{x}\]

L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
\[\lim_{x \rightarrow 0}\frac{sinx}{x}=\lim_{x \rightarrow 0}\frac{\frac{d}{dx}sinx}{\frac{d}{dx}x}\]


d/dx sinx=cosx


d/dx x=1
\[\lim_{x \rightarrow 0}\frac{sinx}{x}=1\]

Answer 4

\[\begin{array}{l} \text{ }\lim_{x\to 0} \, \cos (x) \\ \text{The limit of }\cos x\text{ as }x\text{ approaches }0\text{ is } \\ \text{ }1 \\\end{array}\]

Answer 5

Actually, one cannot use in principle L'Hospital rule in this case. This limit being 1 is used to show that
the derivative of sin(x) is cos(x).

Answer 6

@eliassaab we use L'hopital's rule when the function has a form of 0/0. So, in this case, the rule is applicable

Answer 7

You are using that the derivative of sin(x) is cos(x). To show that the derivative of sin(x) is cos(x) we use the fact that limit is 1.
It works, but in principal is not right to use L'Hospital's rule in this case.

Answer 8

@FoolForMath you might want to say something.

Answer 9

See
http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions
How they prove geometrically that the limit in question is 1 and then prove that the derivative of sin(x) is cos(x)

Answer 10

L'Hospital's Rule is applicable.

Answer 11

I think this problem is more of a verification that L'Hospital's rule works for this indeterminate case. But as proving the limit as of sinx/x as x goes to zero, i'd have to go with the doc.... but thats just me...

Answer 12

in other words, this is the fact that is needed to prove that the derivative of sinx is cosx.

Answer 13

the arc length with angle subtended x = (x)(1) = x
the side of triangle corresponding to the angle x= sin x
as you can see from the drawing, when x is getting smaller, the arc length approaches the length of the triangle side sin x. therefore sin x /x approaches 1 as x approaches 0