Find the indicated limit

Question

2x

limx→∞ (¯¯¯¯¯¯¯¯)

√x² + 1

Answer 1

I'll give you two methods:

1 (the logical way): For large values of "x", √(x² + 1) becomes essentially √x² because, when "x" gets large, the fact that you're adding 1 to it becomes of no consequence. So, for large values of "x", the denominator becomes "x" (√x² = x). So you can cancel an "x" on top and bottom giving you and answer of "2"

2 (the Algebra way):

√(x² + 1) = √x²(1 + 1/x²) = x√(1 + 1/x²) so the expression becomes:

lim (x →∞) (2x/(x√(1 + 1/x²) ) then, canceling the "x's" we get:

lim (x →∞) (2/(√(1 + 1/x²) )

As x →∞, 1/x² approaches zero leaving 2/√1 = 2/1 = 2