3-xsquare<=g(x)<=3secx for all x find lim xtends to0g(x)

To find the limit of the function g(x) as x approaches 0, we can evaluate the left and right limits separately and check if they are equal. Since the expression for g(x) is given as 3 - x^2 ≤ g(x) ≤ 3sec(x), let's handle it step by step.

First, let's consider the left-hand limit (LHL) as x approaches 0. In this case, we need to evaluate the expression for g(x) when x is very close to, but less than, 0.

For the lower bound, g(x) can be at its smallest when x = 0. Therefore, g(x) is greater than or equal to 3 - (0)^2, which simplifies to g(x) ≥ 3.

Next, for the upper bound, sec(x) represents the reciprocal of cosine(x). The maximum value of sec(x) occurs when cos(x) is at its minimum (i.e., -1 or 1), and its value is 1. Therefore, g(x) is smaller than or equal to 3 times sec(x), which simplifies to g(x) ≤ 3.

Combining the two inequalities, we have 3 ≥ g(x) ≥ 3.

Now, let's consider the right-hand limit (RHL) as x approaches 0. In this case, we need to evaluate the expression for g(x) when x is very close to, but greater than, 0.

Using similar reasoning, we find that g(x) is greater than or equal to 3 - (0)^2, which simplifies to g(x) ≥ 3, and g(x) is smaller than or equal to 3 times sec(x), which simplifies to g(x) ≤ 3.

Once again, combining the two inequalities, we have 3 ≥ g(x) ≥ 3.

Since the left-hand limit and the right-hand limit are both equal to 3, we can conclude that the limit of g(x) as x approaches 0 is 3.