QuestionThe function $\frac{|x-2|}{x-2}$ has jump discontinuity at $x=2$ . Select one: True False IVR_TEAM

Studdy Solution

STEP 1

1. The function $f(x) = \frac{|x-2|}{x-2}$ is defined for $x \neq 2$ .
2. A jump discontinuity occurs when the left-hand limit and the right-hand limit of a function at a point exist and are finite, but are not equal to each other.

STEP 2

1. Determine the domain of the function.
2. Calculate the left-hand limit as $x \to 2^-$ .
3. Calculate the right-hand limit as $x \to 2^+$ .
4. Compare the left-hand and right-hand limits to determine if there is a jump discontinuity.

STEP 3

The function $f(x) = \frac{|x-2|}{x-2}$ is defined for all $x \neq 2$ because the denominator becomes zero at $x = 2$ .

STEP 4

Calculate the left-hand limit as $x \to 2^-$ :
For $x < 2$ , $|x-2| = -(x-2)$ . Therefore, the function becomes:
$f(x) = \frac{-(x-2)}{x-2} = -1$
Thus, the left-hand limit is:
$\lim_{x \to 2^-} f(x) = -1$

STEP 5

Calculate the right-hand limit as $x \to 2^+$ :
For $x > 2$ , $|x-2| = x-2$ . Therefore, the function becomes:
$f(x) = \frac{x-2}{x-2} = 1$
Thus, the right-hand limit is:
$\lim_{x \to 2^+} f(x) = 1$

STEP 6

Compare the left-hand and right-hand limits:
The left-hand limit is $-1$ and the right-hand limit is $1$ . Since these limits are not equal, there is a jump discontinuity at $x = 2$ .
The statement is $\boxed{\text{True}}$ .

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