A jump discontinuity occurs when the left-hand limit and right-hand limit both exist but are unequal
A removable discontinuity can be eliminated by redefining the function value at that point.
True
The function f(x)=x−2x2−4 has a removable discontinuity at x=2 because the limit exists, but the function is undefined
Jump discontinuities are removable by redefining the function value.
False
An infinite discontinuity occurs when the function approaches positive or negative infinity
An infinite discontinuity occurs when the function approaches positive or negative infinity at a point.
True
Jump discontinuities have different left-hand and right-hand limits
The key distinction of a jump discontinuity is that the function value is defined but unequal to the limits
The function f(x) = \frac{1}{x - 2}</latex> has an infinite discontinuity at x=2 because it approaches positive or negative infinity
A discontinuity in a function occurs at a point where the function is not continuous
In a jump discontinuity, the function value is defined but not equal to the limits
Jump discontinuities are characterized by a missing or incorrect function value.
False
A jump discontinuity occurs when the left-hand limit and the right-hand limit both exist but are unequal
The function value at a jump discontinuity matches the left-hand limit.
False
A removable discontinuity occurs when the limit exists, but the function value is either undefined or does not match the limit
What can be done to "remove" a removable discontinuity?
Redefine the function value
When does a jump discontinuity occur?
Unequal left and right limits
What happens to the function value at an infinite discontinuity?
It becomes undefined
At a removable discontinuity, the limit of the function exists.
True
Jump discontinuities have different left-hand and right-hand limits
Redefining the function value at a removable discontinuity makes the function continuous at that point.
True
The function f(x)= \begin{cases} x + 1 & \text{if } x \leq 1 \\ 2x & \text{if } x > 1 \end{cases} does not have a jump discontinuity at x=1 because the left and right limits are equal
What is the key characteristic of a jump discontinuity in terms of limits?
Left and right limits are unequal
Match the type of discontinuity with its characteristics:
Removable ↔️ Limit exists, function value incorrect
Jump ↔️ Left and right limits unequal
Infinite ↔️ Function approaches infinity
Steps to remove a removable discontinuity in f(x) = \frac{x^{2} - 4}{x - 2}</latex> at x=2
1️⃣ Factor the numerator
2️⃣ Simplify the expression
3️⃣ Find the limit as x approaches 2
4️⃣ Redefine f(2) to equal the limit
Infinite discontinuities are common in functions with vertical asymptotes.
True
Match the type of discontinuity with its example:
Removable ↔️ f(x)=x−2x2−4 at x=2
Jump ↔️ f(x)= \begin{cases} x + 1 & \text{if } x \leq 1 \\ 2x & \text{if } x > 1 \end{cases} at x=1