Applied Mathematics-IB

Question

Given that $7x\leq f\left ( x \right )\leq 3x^{2}+2$ for all determine the value of $\lim x\rightarrow 0 f(x)$

Answer 1

For the given problem, consider the left-hand and right-hand limits.

 

Left_Hand_Limit = 0,   as   7x ≤ f(x)   i.e   lim(x→0-) 7x = 0.

Right_Hand_Limit = 2,   as   f(x) ≤ 3x^2+2   i.e.   lim(x→0+) 3x^2+2 = 2. 

 

Since, the Left_Hand_Limit ≠ Right_Hand_Limit

Therefore, we have the inequality 0 <= lim(x→0) f(x) <= 2.

 

This means that the limit of f(x) as x approaches 0 exists and is bounded between 0 and 2. However, we cannot determine the exact value of the limit from the given inequality.