Limits at Infinity, Part I — Question 5

Question 5

Determine the limit: $$\lim_{x \to \infty} \frac{5x^4 - 3x^2 + x + 1}{2x^3 + 4x - 7}$$

Question 5 - Solution

We are given: $$\lim_{x \to \infty} \frac{5x^4 - 3x^2 + x + 1}{2x^3 + 4x - 7}$$

Step 1: Compare the degrees of the numerator and denominator.

Degree of numerator: 4 , Degree of denominator: 3

Since the degree of the numerator is greater than the degree of the denominator, the limit is infinite (the function diverges).

Step 2: Determine the sign of the infinity.

As $x \to \infty$, both numerator and denominator are positive: $$\frac{5x^4 + \text{smaller terms}}{2x^3 + \text{smaller terms}} > 0$$

So the limit tends to: $$\boxed{+\infty}$$