Limits and continuity: FAQ

A limit is the value that a function approaches as the input approaches a given value. For example, as we get closer and closer to $x=2$‍ on the function $f(x)={\displaystyle \frac{1}{x}}$‍, the output seems to be getting closer and closer to $0.5$‍. So we say that the limit of $f$‍ as $x$‍ approaches $2$‍ is $0.5$‍.

When we estimate limits from graphs, we look for trends in the function's behavior as the input approaches a certain value. With a table, we look at the function's output as the input values get closer and closer to the value we're interested in. Both methods can help us make educated guesses about what the limit is, but neither is as precise as using algebraic properties of limits.

Continuity refers to a function that doesn't have any "jumps" or "breaks" in it. A function is continuous at a point if the limit as we approach that point from both the left and the right is the same.

Asymptotes are lines that a function approaches, but never quite touches. A vertical asymptote is a line that the function approaches as the input approaches a given value, while a horizontal asymptote is a line that the function approaches as the input goes to infinity or negative infinity.

The squeeze theorem is a way of finding the limit of a function by "sandwiching" it between two other functions whose limits we know.

Limits and continuity are fundamental concepts in calculus, which has countless applications in the real world. Calculus is used in physics, engineering, economics, and more. In particular, limits and continuity can help us understand the behavior of a function over a given interval, and can help us make predictions about how it will behave in the future.