May someone answer this question for me? Thank you! I appreciate it.
a.) Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.
f(x)=(2x+3)/(x-2), (2, ∞)
To show that the function f(x) = (2x + 3)/(x - 2) is continuous on the interval (2, ∞), we need to demonstrate that the function satisfies the definition of continuity and the properties of limits.
The definition of continuity states that a function f(x) is continuous at a point c if three conditions are met:
1. The function f(x) is defined at c.
2. The limit of f(x) as x approaches c exists.
3. The limit of f(x) as x approaches c is equal to f(c).
Let's break down these steps one by one:
1. The function f(x) = (2x + 3)/(x - 2) is defined at all points in the interval (2, ∞), so the first condition is satisfied.
2. To show that the limit of f(x) as x approaches c exists, we need to evaluate the limit. In this case, we want to find the limit as x approaches 2 from the right (since the function is defined for x larger than 2). We can use direct substitution to evaluate the limit:
lim(x→2+) (2x + 3)/(x - 2)
Plugging in x = 2 directly would result in division by zero, which is undefined. However, we can use the properties of limits to simplify the expression.
Since the numerator and denominator are both polynomials, we can factor them. The numerator can be factored as 2x + 3 = 2(x + 3/2), and the denominator is already in factored form as (x - 2). This allows us to cancel out the common factor (x - 2):
lim(x→2+) 2(x + 3/2)/(x - 2)
lim(x→2+) 2(x + 3/2)/(x - 2) * (1/(x - 2))
lim(x→2+) 2(x + 3/2)/((x - 2)(x - 2))
lim(x→2+) 2(x + 3/2)/(x^2 - 4x + 4)
Now, we can substitute x = 2 into the simplified expression:
2(2 + 3/2)/(2^2 - 4(2) + 4)
2(2 + 3/2)/(4 - 8 + 4)
2(2 + 3/2)/0
As we can see, we obtained 0 in the denominator, which indicates an undefined value. Hence, the limit as x approaches 2 from the right does not exist, and the second condition for continuity is not satisfied.
Therefore, we can conclude that the function f(x) = (2x + 3)/(x - 2) is not continuous on the interval (2, ∞).