Evaluate the following limits

Evaluate the following limits:

1.) Limit as x approaches infinity of (x - √(x^2 + 1))

2.) Limit as x approaches infinity of (x - √(x^2 + 4x))

1) lim(x→∞) (x - √(x^2 + 4))
= lim(x→∞) (x - √(x^2 + 4)) (x + √(x^2 + 4))/(x + √(x^2 + 4)), via conjugates
= lim(x→∞) (x^2 - (x^2 + 4))/(x + √(x^2 + 4))
= lim(x→∞) -4/(x + √(x^2 + 4))
= 0.

2) lim(x→∞) (x - √(x^2 + 4x))
= lim(x→∞) (x - √(x^2 + 4x)) (x + √(x^2 + 4x))/(x + √(x^2 + 4x)), via conjugates
= lim(x→∞) (x^2 - (x^2 + 4x))/(x + √(x^2 + 4x))
= lim(x→∞) -4x/(x + √(x^2 + 4))

Now divide each term by x = √x^2:
lim(x→∞) -4/(1 + √(1 + 4/x^2))
= -4/(1 + 1)
= -2.

I hope this helps!

lim as x--> infinity (x-sqrt(x^2+1))
infinity^2 +1 is always infinity
and sort of infinity is infinity
then we left from the x out side that lead us to
infinity - infinity = 0
then the limit as x goes to infinity (x-sqrt(x^2+1)) =0

or you can solve that by using the rationalize and then do the limit of a quotient with L'Hospital rule
u will end up with the same

and for the second one the limits is -2

both 0