Differential Calculus Flashcards
What is a limit in calculus?
In mathematics, a limit is the value that a function approaches as the input approaches some value.
Can a general limit be different from both sides?
No. When a limit doesn’t approach the same value from both sides, we say that the limit doesn’t exist.
Does the function value at a specific input have to be equal to the limit value?
No. The limit as x approaches a value can be different from the function at that x value.
Does a limit that approaches infinity exist?
No. Infinity is not a number. In this situation we say the limit does not exist.
Does a limit that approaches infinity exist?
No. Infinity is not a number. In this situation we say the limit does not exist.
What does “infinitely close” mean in the context of a limit?
That no matter how close we want to get to the limit, there’s an x-value that will get us there.
When are there holes in graphs?
Holes in graphs happen with rational functions, which become undefined when their denominators are zero.
When a function is defined for some x-value, does a limit always exist?
when the function is defined for some x-value, that doesn’t mean that the limit necessarily exists. Just like an earlier example, this graph shows the sort of thing that can happen when we’re working with piecewise functions.
Just because a function is undefined for some x-value does that mean there’s no limit?
No. There can be a limit even if the function is undefined for that x-value.
What is a one-sided limit?
A one-sided limit is the value the function approaches as the x-values approach the limit from one side only.
What is an unbounded limit?
If the limit the graph is approaching is infinity, the limit is unbounded.
What are the 5 most common mistakes when creating tables to estimate limits?
- Assuming the function value is the limit value
- Not getting infinitely close
- Not approaching from both sides
- Assuming “left side” means “negative”
- Thinking a limit value is always an integer
What is the exercise of formally defining a limit about?
- You tell me how close you want f(x) to be to L (give positive number ϵ).
- I will find you another positive number δ where if x is within δ of c, then f(x) will be within ϵ of L.
How do you formally define a limit?
We want to make the inequality describing the distance delta look like the inequality describing the distance epsilon.
What is the lim_x->c [f(x) + g(x)] equal to?
It is equal to the lim_x->c f(x) + lim_x->c g(x).
