Question

Prof says answer is 5 to the following:
lim as x goes to 0 of (sin 5x)/x. is he right and how does it work?

Answer 1

hi big nose

Answer 2

let u = 5x

Answer 3

as x ->0 , u -> 5*0 = 0 , and u = 5x so u/5 = x

now we have lim u-> 0 sin u / ( u/5) = lim 5 * sin u/u

Answer 4

we know that lim sin x / x as x goes to zero is 1, this is a known fact or theorem. you can prove it seperately using squeeze theorem or by geometry

Answer 5

so lim 5 * sin u/u = 5 * lim sin u/u = 5 * 1

Answer 6

\[\lim_{x \rightarrow 0} \sin(5x)/x\]
produces the indeterminate form 0/0, so you can use L'Hopital's rule that:
\[\lim_{x \rightarrow a}f(x)/g(x) = \lim_{x \rightarrow a} f \prime (x) / g \prime (x)\]
(keep in mind that you can only use this if the original limit produces 0/0 or infinity/infinity!). So, applying the rule gives us:
\[\lim_{x \rightarrow 0}5\cos(x)\]
cos(0) is one, so the answer is 5.