lim (x+3)/(√(9x^2-5x))

x-> infinity

This limit can be reduced to this

lim (x+3)/(|x|√(9-5/x))

x-> infinity

All you have to remember is that √x^2 = |x|

So if you factor out x^2 from under the square root you get

lim (x+3)/(√(x^2(9-5/x))

x-> infinity

Which is the same thing as

lim (x+3)/(|x|√(9-5/x))

x-> infinity

x --> infinity so x is always +

Divide everything by x

lim (1+3/x )/(√(9-5/x)) =1/3

x-> infinity

2)

lim ((√(x^2+1))-x)

x ->infinity

divide everything by x like the first question

lim ((√(1+1/x^2))-1)/(1/x)

x ->infinity

use a substitution t = 1/x

lim ((√(1+t^2)-1)/(t)

t->0

You can see that this is really the definition of a derivative just with t not h.

lim [f(x+h) - f(h)]/ h

h --> 0

f(x) = √(1+t^2)

f '(x) = 2t /2(√(1+t^2)

f ' (0) = 0 /2 = 0

So the limit = 0