Question

Let f(x) = { x² - 9/ x -3 , x does not =3

3, x = 3​

Which of the following statements are true?
I. lim f(x) exists
x→3
II.f(3) exists
III. f(x) is continuous at x=3

Answer 1

The limit of f(x) as x approaches 3 exists and f(3) is defined. However, the function is not continuous at x = 3 because the limit of f(x) as x approaches 3 is not equal to f(3). All steps analyzing the limit and continuity are explained with relevant calculations.

Step-by-step explanation:

The question asks to evaluate the statements about the function f(x):
f(x) = { x² - 9/ x -3 , x does not =3 3, x = 3}

For statement I, to find if the limit of f(x) as x approaches 3 exists, we must analyze the behavior of f(x) as x gets arbitrarily close to 3 from both sides. If the left-hand limit and right-hand limit as x approaches 3 are equal, then the limit exists.

In statement II, f(3) exists because the function is explicitly defined at x = 3.

For statement III, continuity at x = 3 involves meeting three conditions: the limit as x approaches 3 exists, f(3) exists, and the limit of f(x) as x approaches 3 is equal to f(3).

To check for continuity:

1. Compute the limit of (x² - 9) / (x - 3) as x approaches 3.
2. Check the value of f(3).
3. Determine if these two values are equal.

Steps to solve:

• Factor the numerator x² - 9 as (x + 3)(x - 3).
• Cancel the (x - 3) in the numerator and denominator.
• The function simplifies to x + 3 when x does not equal 3.
• Evaluate the limit of x + 3 as x approaches 3, which is 6.
• The value of f(3) is given as 3.
• Since the limit of f(x) as x approaches 3 is 6 and f(3) = 3, the function is not continuous at x = 3.

Therefore, statement I is true, statement II is true, and statement III is false because the limit of f(x) as x approaches 3 does not equal f(3).

Answer 2

f(x) = (x² - 9)/(x - 3)

= (x - 3)(x + 3)/(x - 3)

= x + 3 (x ≠ 3)

f(x) = 3 (x = 3)

Statements I and II are true.