If 6x ≤ g(x) ≤ 3x4 − 3x2 + 6 for all x, evaluate lim x→1 g(x).

Question

If 6x ≤ g(x) ≤ 3x⁴ − 3x² + 6 for all x, evaluate lim x→1 g(x).

Answer 1

By taking the limit of all three components of the inequality, we see that the limit as x approaches 1 of 6x is 6. Moreover, we see that the limit as x approaches 1 of 3x⁴ - 3x² + 6 is also 6. Because g(x) is between these two functions for all x, we know by the squeeze theorem that the limit as x approaches 1 of g(x) must also be 6.