Limits and Continuity Limits A limit is the value a function approaches as its input approaches a specific value. We denote the limit of a function f...
Limits
A limit is the value a function approaches as its input approaches a specific value. We denote the limit of a function f(x) as x approaches a value a by lim_(x->a) f(x). The limit tells us what the function approaches as x gets infinitely close to a.
Examples:
lim_(x->0) x = 0 because the function approaches 0 as x approaches 0.
lim_(x-> infinity) x^2 = infinity since the function approaches infinity as x approaches infinity.
lim_(x->2) (x+1) = 3 since the function approaches 3 as x approaches 2.
Continuity
A function is continuous at a point a if the limit of the function as x approaches a is equal to the value of the function at that point. In other words, the function takes on the same value at a and the limit is equal to the function value.
Examples:
The function f(x) = x^2 is continuous for all real values of x because the limit of x^2 as x approaches a is equal to the function value at x = a (a = 0).
The function f(x) = 1/x is continuous for all real values of x except x = 0 because the limit of 1/x as x approaches 0 is undefined.
The function f(x) = sin(x) is not continuous at x = π/2 because the limit of sin(x) as x approaches π/2 is not equal to the function value at x = π/2.
Key Differences:
Limits tell us what the function approaches as its input approaches a specific value, while continuity tells us whether the function takes the same value at a point.
Limits can be calculated directly, while continuity requires evaluating the limit.
Continuity is a stronger condition than limits. A function can be continuous at a point without having a limit at that point