If
12x − 52 ≤ f(x) ≤ x2 + 4x − 36
determine lim x→ 4 f(x) =
What theorem did you use to arrive at your answer?
on the left, as x>4, 48-52
On the right, as x>4, 16+16-36
so
-4<=f(x)<=-4
so, squeezed from the right, squeezed from the left, lim f(x) x>4 = -4
To determine the limit, we need to evaluate the function as x approaches 4. However, before we proceed, let's simplify the inequality first:
12x - 52 ≤ f(x) ≤ x^2 + 4x - 36
Now, let's take the limit as x approaches 4 for each term of the inequality:
lim x→4 (12x - 52) = 12(4) - 52 = 48 - 52 = -4
lim x→4 (x^2 + 4x - 36) = (4)^2 + 4(4) - 36 = 16 + 16 - 36 = -4
Since the limit of both terms is equal to -4, we can conclude that the limit of f(x) as x approaches 4 is also -4.
The theorem used to arrive at this answer is the Squeeze or Sandwich theorem. This theorem states that if two functions, g(x) and h(x), both squeeze a third function f(x) between them for values of x close to a particular point (in this case, x = 4), then the limit of f(x) as x approaches that point will be the same as the limits of g(x) and h(x) at that point.