Limits

[Reallyman]

Limits are fun. Ok so I already know the answer but i am curious as to what type of magic occurred between the lines because my teacher did not explain or my tutors.
So,
Lim Sin(6x)/x = 6 BTW
X approachs 0

If anyone could break this down to a mentally handicap person's level I think then I may benefit. BTW I accept the mathematical fact that,

Lim Sin(x)/x= 1 I just need the break down. Any help will be greatly appreciated.

 

[lookagain]

Lim sin(6x)/x = 6 BTW
x approaches 0


Lim sin(x)/x = 1 \(\displaystyle \ as \ x \ approaches \ 0.\)

\(\displaystyle Using \ \lim_{x \to 0}\bigg[\frac{sin(x)}{x}\bigg] = 1:\)


\(\displaystyle Then \ the \ \lim_{x \to 0}\bigg[\frac{sin(6x)}{6x}\bigg] = 1\)

This equation becomes:


\(\displaystyle \frac{1}{6} \bigg \{ \lim_{x \to 0}\bigg[\frac{sin(6x)}{x}\bigg] \bigg \} \ = \ 1\)


Then multiply each side by \(\displaystyle 6.\)


There are more steps that could be written to justify changing from the
second to the last equation into the last equation.

 

[DrSteve]

You can also replace \(\displaystyle x\rightarrow 0\) by \(\displaystyle 6x\rightarrow 0\) because it is clear that \(\displaystyle x\) approaches \(\displaystyle 0\) if and only if \(\displaystyle 6x\) does. You can then make the substitution \(\displaystyle u=6x\) to make the new limit look identical to the one that you are ok assuming to be true.

 

[Reallyman]

Thank You very very much fellows. The help was right on the money!