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Lesson Example Discussion Quiz: Class Homework

Step-4

Title: Limits and continuity

Grade: 1300-a Lesson: S2-L4

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
1	Problem	Determine if the function \$f(x) = (x^2 - 4)/(x - 2)\$ is continuous at x = 2.
2	Step	To check the continuity of the function f(x) at x = 2, we need to evaluate the following three conditions. 1. The function must be defined at that point. 2. The limit of the function as x approaches that point must exist. 3. The value of the function at that point must be equal to the limit.
3	Step	The function is not defined at x = 2 due to division by zero (denominator becomes zero). However, it can be simplified by factoring the numerator	\$f(x) = (x^2 - 4)/(x - 2)\$
4	Hint	Let’s first check if the function is defined at x = 2. By plugging in x = 2 into the function, we can determine	\$ f(2) = (2^2 - 4)/(2 - 2) = 0/0\$
5	Step	The function is not defined for x = 2.
6	Step	Next, we need to find the limit of the function as x approaches 2. Taking the limit as x approaches 2, we have:	\$\lim_{x \to 2} {(x^2 - 4)/(x - 2)} = \lim_{x \to 2} {((x + 2)(x - 2))/(x - 2)} \$
7	Hint	Factoring out the common factor (x - 2)	\$\lim_{x \to 2} {(x^2 - 4)/(x - 2)} = \lim_{x \to 2} {(x + 2)} \$
8	Clue	Substituting x = 2, then the limit exists and is equal to 4	\$\lim_{x \to 2} {(x^2 - 4)/(x - 2)} = 2 + 2 = 4\$
9	Step	The value of the function at x = 2 must be equal to the limit. Since the function is undefined at x = 2, the value of the function cannot be compared to the limit.
10	Step	As the function is not defined at x = 2, it fails the first condition and is therefore not continuous at x = 2.
11	Choice.A	This statement is unclear about the function’s continuity at x = 2, and it provides an unrelated value of \$1/2\$	\$ 1/2\$
12	Choice.B	The function is discontinuous at x = 2 due to lack of definition with unequal limits on either side of x = 2	Discontinuous
13	Choice.C	This statement is unclear about the function’s continuity at x = 2, and it provides an unrelated value of 4	4
14	Choice.D	This option suggests that the function is continuous at x = 2. However, as we’ve determined earlier, the function is discontinuous at x = 2 due to division by zero	Continuous
15	Answer	Option	B
16	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Determine if the function \$f(x) = (x^2 - 4)/(x - 2)\$ is continuous at x = 2.

Step

To check the continuity of the function f(x) at x = 2, we need to evaluate the following three conditions.
1. The function must be defined at that point.
2. The limit of the function as x approaches that point must exist.
3. The value of the function at that point must be equal to the limit.

Step

The function is not defined at x = 2 due to division by zero (denominator becomes zero). However, it can be simplified by factoring the numerator

\$f(x) = (x^2 - 4)/(x - 2)\$

Hint

Let’s first check if the function is defined at x = 2. By plugging in x = 2 into the function, we can determine

\$ f(2) = (2^2 - 4)/(2 - 2) = 0/0\$

Step

The function is not defined for x = 2.

Step

Next, we need to find the limit of the function as x approaches 2. Taking the limit as x approaches 2, we have:

\$\lim_{x \to 2} {(x^2 - 4)/(x - 2)} = \lim_{x \to 2} {((x + 2)(x - 2))/(x - 2)} \$

Hint

Factoring out the common factor (x - 2)

\$\lim_{x \to 2} {(x^2 - 4)/(x - 2)} = \lim_{x \to 2} {(x + 2)} \$

Clue

Substituting x = 2, then the limit exists and is equal to 4

\$\lim_{x \to 2} {(x^2 - 4)/(x - 2)} = 2 + 2 = 4\$

Step

The value of the function at x = 2 must be equal to the limit. Since the function is undefined at x = 2, the value of the function cannot be compared to the limit.

Step

As the function is not defined at x = 2, it fails the first condition and is therefore not continuous at x = 2.

Choice.A

This statement is unclear about the function’s continuity at x = 2, and it provides an unrelated value of \$1/2\$

\$ 1/2\$

Choice.B

The function is discontinuous at x = 2 due to lack of definition with unequal limits on either side of x = 2

Discontinuous

Choice.C

This statement is unclear about the function’s continuity at x = 2, and it provides an unrelated value of 4

Choice.D

This option suggests that the function is continuous at x = 2. However, as we’ve determined earlier, the function is discontinuous at x = 2 due to division by zero

Continuous

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?