What is #lim x->∞# of #(7x+1)/(11x-6)#?

1 Answer
Jul 18, 2016

#lim_(x->∞) (7x+1)/(11x-6) = 7/11#

Explanation:

This limit is known as an infinite limit. For polynomials, we can divide both the numerator and the denominator by the highest power (in this case, #x# is the highest power).

#lim_(x->∞) (7x+1)/(11x-6) = lim_(x->∞) ((7cancel(x))/(cancel(x))+1/(x))/((11cancel(x))/(cancel(x))-6/(x)) = lim_(x->∞) (7+1/x)/(11-6/x)#

We can use the fact that #lim_(x->∞) 1/x = 0#, so evaluating the limit certainly becomes easier now.

#lim_(x->∞) (7+cancel(1/x))/(11-cancel(6/x)) = 7/11 ≈ 0.6363#

Even if this may not be as clear, we can also verify this by graphing.

Graph of #f(x) = (7x+1)/(11x-6)#:

graph{(7x+1)/(11x-6) [-2.34, 5.457, -0.592, 3.306]}