Question

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

Answer 1

`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]}