Limits and Continuity Limits A limit is the value a function approaches as the input approaches a specific value. In other words, it tells us what th...
Limits
A limit is the value a function approaches as the input approaches a specific value. In other words, it tells us what the function does as its input gets closer and closer to the given value.
Formal definition:
For a function f(x) and a fixed value a, the limit of f(x) as x approaches a is L if:
lim_(x->a) f(x) = L
Examples:
lim_(x->0) x^2 = 0 because the function approaches 0 as x gets closer and closer to 0.
lim_(x->10) (x+100) = 120 because the function approaches 120 as x gets closer to 100.
lim_(x->1/2) (x+1/2)^2 = 1 because the function approaches 1 as x gets closer to 1/2.
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 a. In other words, the function can be differentiated at a.
Formal definition:
A function f(x) is continuous at a point a if:
lim_(x->a) f(x) = f(a)
Examples:
f(x) = x^2 is continuous for all real numbers because the limit of f(x) as x approaches a is equal to the value of f(a) (which is 0).
f(x) = sin(x) is continuous for all real numbers because the limit of f(x) as x approaches a is equal to the value of f(a) (which is 0).
f(x) = 1/x is not continuous for x = 0 because the limit of f(x) as x approaches 0 is undefined.
Key differences:
A limit tells us what the function approaches as x gets closer to a, while a continuous function has a specific value at a.
Limits can be found by evaluating the function at a specific value, while continuity requires finding the limit.
Limits are used to determine the derivatives and integrals of functions, which are important concepts in calculus