Find the indicated limit, if it exists. (If an answer does not exist, enter DNE.)

Question

Find the indicated limit, if it exists. (If an answer does not exist, enter DNE.)

lim

x→∞ (24x^4 − 3x^2 + 4)

(8x^4 + x^3 + x^2 + x + 4)

Answer 1

There are different ways to approach this problem, here's one:

The 'dominant' term (as x approaches ∞) in the numerator is x^4, the dominant term in denominator is also x^4, so multiply the numerator and denominator by 1/x^4:

lim_{x \to ∞} [ (1/x^4) ( 24x^4-3x^2+4)/ (1/x^4)(8x^4+x^3+x^2+x+4) ]=

lim_{x \to ∞} ( 24 -3/x^2+4/x^4) / (8 +1/x+1/x^2+1/x^3+4/x^4)

The advantage of this equivalent form is that the limits of the numerator and denominator exist individually, and so we can just divide them:

lim_{x \to ∞} ( 24 -3/x^2+4/x^4) / (8 +1/x+1/x^2+1/x^3+4/x^4) = 24/8 = 3

(since as x approaches infinity, -3/x^2, for instance, goes to 0, etc.)