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(without using l'hopital's rule) how to do you prove that:

lim x->0 (sinx)/x=1? 

lim x->0 (tan3x)/x =3?

lim x->0 (1-cosx)x? 

 Jan 30, 2015

Best Answer 

 #3
avatar+111470 
+10

For the second one, we have

tan(3x)/x  = (sin3x)/x * 1/cos3x = (3sin3x/3x) * 1/(cos 3x) 

Let 3x = Θ

And  using the fact that sinΘ/Θ = 1, and cosΘ = 1   as Θ→ 0..... we have

(3sinΘ/Θ) * 1/cosΘ = (3)(sinΘ/Θ) * 1/cosΘ =

[3(1)] / [1/1]  = 3/1  = 3  

 

    

 Jan 30, 2015
 #1
avatar+25574 
+10

(without using l'hopital's rule) how to do you prove that:

lim x->0 (sinx)/x=1 ? 

$$\small{\text{
$
\boxed{ \lim\limits_{x\to0}
\left(
\dfrac{ \sin{(x)} } { x } \right)
\qquad \sin (x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!} = \frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}\mp\dotsb
} % boxed
$
}}$\\\\\\$
\small{\text{
$
\lim\limits_{x\to0}\left(
\dfrac{ \sin{(x)} } { x } \right)
=
\lim\limits_{x\to0}\left(
\dfrac{
\dfrac{x}{1!}-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}\mp\dotsb
} { x } \right)
=
\lim\limits_{x\to0}\left(
\dfrac{
x \left( \dfrac{1}{1!}-\dfrac{x^2}{3!}+\dfrac{x^4}{5!}\mp\dotsb \right)
} { x } \right)
$
}}$\\\\\\$
\small{\text{
$
=
\lim\limits_{x\to0}\left(
\dfrac{1}{1!}-\dfrac{x^2}{3!}+\dfrac{x^4}{5!}\mp\dotsb \right)= \dfrac{1}{1!} = \textcolor[rgb]{1,0,0}{1}
$
}}$$

 Jan 30, 2015
 #2
avatar+25574 
+5

(without using l'hopital's rule) how to do you prove that:

lim x->0 (tan3x)/x =3 ? 

$$\small{\text{
$
\boxed{ \lim\limits_{x\to0}
\left(
\dfrac{ \tan{(3x)} } { x } \right)
\qquad \tan{ (3x) } &= (3x)+\dfrac13 (3x)^3+\dfrac{2}{15}(3x)^5+\dfrac{17}{315}(3x)^7+\dotsb
} % boxed
$
}}$\\\\\\$
\small{\text{
$
\lim\limits_{x\to0}\left(
\dfrac{ \tan{(3x)} } { x } \right)
=
\lim\limits_{x\to0}\left(
\dfrac{
(3x)+\dfrac13 (3x)^3+\dfrac{2}{15}(3x)^5+\dfrac{17}{315}(3x)^7+\dotsb
} { x } \right)
=
\lim\limits_{x\to0}\left(
\dfrac{
x \left( 3+\dfrac13 3^3x^2+\dfrac{2}{15}3^5x^4+\dfrac{17}{315}3^7x^6+\dotsb \right)
} { x } \right)
$
}}$\\\\\\$
\small{\text{
$
=
\lim\limits_{x\to0}\left(
3+\dfrac13 3^3x^2+\dfrac{2}{15}3^5x^4+\dfrac{17}{315}3^7x^6+\dotsb \right)= \textcolor[rgb]{1,0,0}{3}
$
}}$$

 Jan 30, 2015
 #3
avatar+111470 
+10
Best Answer

For the second one, we have

tan(3x)/x  = (sin3x)/x * 1/cos3x = (3sin3x/3x) * 1/(cos 3x) 

Let 3x = Θ

And  using the fact that sinΘ/Θ = 1, and cosΘ = 1   as Θ→ 0..... we have

(3sinΘ/Θ) * 1/cosΘ = (3)(sinΘ/Θ) * 1/cosΘ =

[3(1)] / [1/1]  = 3/1  = 3  

 

    

CPhill Jan 30, 2015

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