Calculus I Flashcards
When does a limit fail to exist due to a jump?
A limit does not exist when the left-hand limit and right-hand limit approach different values at a point. This occurs in piecewise functions with a ‘jump’ discontinuity.
What happens if a function approaches infinity?
If the function grows to positive or negative infinity at a certain point, the limit does not exist. This happens near vertical asymptotes.
How does oscillation prevent a limit from existing?
If a function oscillates infinitely (e.g., sin(1/x) near x = 0), it does not settle on a single value, so the limit does not exist.
How do you identify a removable discontinuity?
A function has a removable discontinuity when there is a hole at a point where the function is undefined, but the limit exists. This happens when a factor cancels in a rational function.
What is a jump discontinuity?
A function has a jump discontinuity if the left-hand and right-hand limits approach different values. This occurs in piecewise functions where the function ‘jumps’ from one value to another.
How do you identify an infinite discontinuity?
If a function approaches positive or negative infinity at a point, there is an infinite discontinuity. This is seen near vertical asymptotes.
What is an oscillatory discontinuity?
A function has an oscillatory discontinuity if it oscillates infinitely near a point and does not settle on a single value. Example: sin(1/x) at x = 0.
