What is the best way to solve problem like these ↓ ? (Limits to infinity)

Problems such as

When #x# approaches positive infinity, find the limit of

#(3x-1)/(sqrt(x^2-6#

#(sqrt(4x^2+4x))/(4x+1)#

I know that you can solve by substituting values of #x# as it gets closer to infinity, but is there a way to solve, like factorising, formula, or a general rule?

1 Answer
Jun 22, 2017

Answer:

As a general rule convert a polynomial #f(x)# to #g(1/x)#
(A) #3# and (B) #1/2#

Explanation:

#Lt_(x->oo)(3x-1)/sqrt(x^2-6)#

  • let us divide numerator and denominator by #x# and we get

#Lt_(x->oo)(3-1/x)/sqrt(1-6/x^2)#

= #3/sqrt1=3#
graph{(3x-1)/sqrt(x^2-6) [2.07, 19.55, -0.37, 8.37]}

#Lt_(x->oo)sqrt(4x^2+4x)/(4x+1)#

  • dividing numerator and denominator by #2x# and we get

#Lt_(x->oo)sqrt(1+2/x)/(2+1/2x)#

= #1/2#
graph{sqrt(4x^2+4x)/(4x+1) [-0.152, 4.218, -0.51, 1.675]}

As a general rule convert a polynomial #f(x)# to #g(1/x)#