TWO examples of each of the following:

two sided limits
and
theorem on limits (SUM, difference, product, quotient, nth root, ).

Two-Sided Limits:

1. Example 1: Let's consider the function f(x) = 2x^2 - 5x + 3. To find the two-sided limit of f(x) as x approaches 2, we need to evaluate the function from both the left and right sides of x = 2. We calculate f(x) for some values slightly less than 2 and slightly greater than 2.

For x-values approaching 2 from the left side (x < 2):
- f(1.9) = 2(1.9)^2 - 5(1.9) + 3 = 7.04
- f(1.99) = 2(1.99)^2 - 5(1.99) + 3 = 7.0396

For x-values approaching 2 from the right side (x > 2):
- f(2.1) = 2(2.1)^2 - 5(2.1) + 3 = 5.04
- f(2.01) = 2(2.01)^2 - 5(2.01) + 3 = 5.0396

Therefore, the two-sided limit of f(x) as x approaches 2 is 5.04 when x approaches 2 from the right side and 7.04 when x approaches 2 from the left side.

2. Example 2: Consider the function g(x) = (x^2 - 4) / (x - 2). Finding the two-sided limit of g(x) as x approaches 2 requires calculating the limit from both the left and right sides.

For x-values approaching 2 from the left side (x < 2):
- g(1.9) = (1.9^2 - 4) / (1.9 - 2) = -3.9
- g(1.99) = (1.99^2 - 4) / (1.99 - 2) = -3.99

For x-values approaching 2 from the right side (x > 2):
- g(2.1) = (2.1^2 - 4) / (2.1 - 2) = 3.9
- g(2.01) = (2.01^2 - 4) / (2.01 - 2) = 3.99

In this case, the two-sided limit of g(x) as x approaches 2 does not exist because the limits from the left and right sides differ. The function has a vertical asymptote at x = 2.

Theorem on Limits (SUM, difference, product, quotient, nth root):
The theorem states that if you have two functions f(x) and g(x), and their individual limits as x approaches a exist, then the following operations can be applied:

1. Sum: The limit of the sum of two functions equals the sum of their limits.
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) + g(x)] = L + M.

2. Difference: The limit of the difference of two functions equals the difference of their limits.
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) - g(x)] = L - M.

3. Product: The limit of the product of two functions equals the product of their limits.
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) * g(x)] = L * M.

4. Quotient: The limit of the quotient of two functions equals the quotient of their limits provided the limit of the denominator is not zero.
- If lim(x->a) f(x) = L, lim(x->a) g(x) = M, and M ≠ 0, then lim(x->a) [f(x) / g(x)] = L / M.

5. Nth Root: The limit of the nth root of a function equals the nth root of its limit.
- If lim(x->a) f(x) = L, and n is a positive integer, then lim(x->a) [f(x)]^(1/n) = L^(1/n).

These theorems allow us to evaluate the limits of more complex functions by breaking them down into simpler components and applying the appropriate operation based on the theorem.