Math  /  Calculus

Questionf(x)=x213x+36x4f(x)=\frac{x^{2}-13 x+36}{x-4} (a) Explain why ff has a removable discontinuity at x=4x=4. (Select all that apply.) f(4)f(4) is undefined. f(4)f(4) and limx4f(x)\lim _{x \rightarrow 4} f(x) are finite, but are not equal. limx4f(x)\lim _{x \rightarrow 4} f(x) does not exists. limx4f(x)\lim _{x \rightarrow 4} f(x) is finite. none of the above

Studdy Solution

STEP 1

1. The given function is f(x)=x213x+36x4 f(x) = \frac{x^2 - 13x + 36}{x - 4} .
2. We need to determine if there is a removable discontinuity at x=4 x = 4 .
3. A removable discontinuity occurs if f(x) f(x) is not defined at x=4 x = 4 but the limit limx4f(x) \lim_{x \to 4} f(x) exists and is finite.

STEP 2

1. Verify if f(4) f(4) is undefined by substituting x=4 x = 4 into the function.
2. Determine if the limit limx4f(x) \lim_{x \to 4} f(x) exists and is finite.
3. Analyze the above information to conclude about the removable discontinuity.

STEP 3

Substitute x=4 x = 4 into f(x) f(x) to see if it is defined.
f(4)=42134+3644 f(4) = \frac{4^2 - 13 \cdot 4 + 36}{4 - 4}
Since the denominator becomes zero, f(4) f(4) is undefined.

STEP 4

Simplify the numerator x213x+36 x^2 - 13x + 36 by factoring.
x213x+36=(x4)(x9) x^2 - 13x + 36 = (x - 4)(x - 9)
Thus, we can rewrite the function as:
f(x)=(x4)(x9)x4 f(x) = \frac{(x - 4)(x - 9)}{x - 4}
For x4 x \neq 4 , this simplifies to:
f(x)=x9 f(x) = x - 9

STEP 5

Determine the limit limx4f(x) \lim_{x \to 4} f(x) using the simplified form of the function.
limx4f(x)=limx4(x9)=49=5 \lim_{x \to 4} f(x) = \lim_{x \to 4} (x - 9) = 4 - 9 = -5
The limit exists and is finite.

STEP 6

Analyze the information to conclude about the removable discontinuity.
1. f(4) f(4) is undefined.
2. limx4f(x) \lim_{x \to 4} f(x) exists and is finite.

Therefore, f(x) f(x) has a removable discontinuity at x=4 x = 4 .
Solution: The correct options are: - f(4) f(4) is undefined. - limx4f(x) \lim_{x \to 4} f(x) is finite.

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