Limits and Continuity Review

Front

What is a limit in calculus?

Back

A limit is a value that a function approaches as the input approaches some value. It is fundamental in defining continuity, derivatives, and integrals.

Front

Define continuity at a point.

Back

A function f(x) is continuous at a point x = a if: 1) f(a) is defined, 2) the limit as x approaches a exists, and 3) the limit equals f(a).

Front

What does it mean if a limit does not exist (DNE)?

Back

A limit does not exist if the function approaches different values from the left and right, or if it approaches infinity.

Front

How do you evaluate limits at infinity for polynomials?

Back

For polynomials, the limit as x approaches infinity is determined by the leading term. If the leading term is positive and odd, the limit is +∞.

Front

Back

The limit does not exist (DNE) because it approaches +∞ from the right and -∞ from the left.

Front

What is the limit of a constant function?

Back

The limit of a constant function as x approaches any value is simply the constant itself.

Front

Explain the concept of one-sided limits.

Back

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side: left-hand limit (x approaches a from the left) and right-hand limit (x approaches a from the right).

Front

Back

The limit is 2, as the leading terms dominate the behavior of the function.

Front

What is the squeeze theorem?

Back

The squeeze theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in some interval, and if the limits of f(x) and h(x) both approach L as x approaches a, then the limit of g(x) also approaches L.

Front

Back

The limit is 2, after factoring and simplifying the expression.