limits

[calculusconfused]

i have several limits which initially result in IND forms but i cannot figure out how to simplify them..

1. lim x=>0+ sqrt(x)*e^sin(pi/x)

2. lim x=>0 ((8x+sin4x)/sin3x)+(xcotx/4)


thank you so much!!!

 

[galactus]

i have several limits which initially result in IND forms but i cannot figure out how to simplify them..

1. lim x=>0+ sqrt(x)*e^sin(pi/x)

2. lim x=>0 ((8x+sin4x)/sin3x)+(xcotx/4)


thank you so much!!!

Is that \(\displaystyle cot(\frac{x}{4})\) or \(\displaystyle \frac{cot(x)}{4}\)?.

 

[calculusconfused]

it is (x*cot(x))/4

 

[galactus]

2.\(\displaystyle \lim_{x\to 0}\frac{8x+sin(4x)}{sin(3x)}+\frac{xcot(x)}{4}\)

\(\displaystyle \lim_{x\to 0}\frac{8x}{sin(3x)}+\lim_{x\to 0}\frac{sin(4x)}{sin(3x)}+\lim_{x\to 0}\frac{xcot(x)}{4}\)

The first two can be found by knowing, near 0, sin(x) behaves like x.

So, we can write \(\displaystyle \lim_{x\to 0}\frac{8x}{sin(3x)}\approx \lim_{x\to 0}\frac{8x}{3x}=\frac{8}{3}\)

and

\(\displaystyle \lim_{x\to 0}\frac{sin(4x)}{sin(3x)}\approx \lim_{x\to 0}\frac{4x}{3x}=\frac{4}{3}\)

The third one can be rewritten as \(\displaystyle \frac{1}{4}\lim_{x\to 0}\frac{x}{tan(x)}\).

The limit x/tan(x) is rather 'famous' and evaluates to 1. Like the famous \(\displaystyle \lim_{x\to 0}\frac{sin(x)}{x}=1\) which is often used as a lemma in other limits. See?. This limit is why we can say that near 0 sin(x) behaves like x. Same with tan(x).

So, we get \(\displaystyle \frac{8}{3}+\frac{4}{3}+\frac{1}{4}=\frac{17}{4}\)

 

[calculusconfused]

thank you very much!!

 

[calculusconfused]

any ideas for the first one?

 

[renegade05]

any ideas for the first one?

Is that:
\(\displaystyle \lim_{x\to 0+}\sqrt{x}\cdot e^{sin(\frac{\pi}{x})}\)??

If so:

\(\displaystyle =\lim_{x\to 0+}\sqrt{x}\cdot \lim_{x\to 0+}e^{sin(\frac{\pi}{x})}\)

\(\displaystyle =0\cdot\lim_{x\to 0+}e^{sin(\frac{\pi}{x})}\)

\(\displaystyle sin(\frac{\pi}{x})\) is approaching infinity or negative infinity so e to infinity or neg infinity will be a very big or very small number, but it does'nt matter since it is getting multiplied by zero.

\(\displaystyle =\lim_{x\to 0+}\sqrt{x}e^{sin(\frac{\pi}{x})}=0\)