lim problems: lim, x->4, [sqrt(x+5)-3]/(x-4), lim, x->

[kpx001]

lim [sqroot(x+5)-3]/(x-4)
x->4

i need help with this problem. I multiplied by the conjugate [sqroot(x+5)+sqroot(x)]/ [sqroot(x+5)+sqroot(x)] and got x+5-5 / x[sqroot(x+5)+sqroot(x)] but i dunno what to do next.
also this problem too.

lim cosx/cotx = cosx/(cosx/sinx)
x->pi/2

i cant split the problem because i just get (cosx)(cosx/sinx) and cant get either 1-cosx or sinx/x

 

[galactus]

#1: Yes, use the conjugate.

\(\displaystyle \L\\\frac{(\sqrt{x+5}-3)}{x-4}\cdot{\frac{(\sqrt{x+5}+3)}{\sqrt{x+5}+3}}\)

=\(\displaystyle \L\\\frac{x-4}{(x-4)(\sqrt{x+5}+3)}=\frac{1}{\sqrt{x+5}+3}\)

Now you have:

\(\displaystyle \L\\\lim_{x\to\4}\frac{1}{\sqrt{x+5}+3}\)

Now, see the limit?.


For the second one. cos(x)/cot(x)=sin(x). That's it.