Question

Question #a0b5b

Answer 1

We use different methods for different kinds of functions.

Explanation:

The function you've asked about is a rational function and there is a cookbook for this kind of function. (Or a spacial tool in out mathematical toolbox. Choose your imagery.)

Find the greatest power of `x` in the denominator.

Divide numerator and denominator by that power of `x`. (Long ago, I was taught to factor it out, and then reduce.)

`lim_(xrarroo)(x+x^3+x^5)/(1+x+x^5) = lim_(xrarroo)(x^5(1/x^4+1/x^2+1))/(x^5(1/x^5+1/x^4+1))`

`= lim_(xrarroo)(1/x^4+1/x^2+1)/(1/x^5+1/x^4+1)`

Now use the fact that for any real number `k` and any positive `n`, `lim_(xrarroo)k/x^n = 0` to get

`= (0+0+1)/(0+0+1) = 1`

Bonus examples

`lim_(xrarroo)(4x^3-2x+7)/(5x^3+4x^2-1) = lim_(xrarroo)(4-2/x^2+7/x^3)/(5+4/x-1/x^3) = 4/5`

`lim_(xrarroo)(x^2+5x-3)/(2x^3+x^2+4) = lim_(xrarroo)(1/x+5/x^2-3/x^3)/(2+1/x+4/x^3) = 0/2 = 0`