The limit of a function f is a tool for investigating the behavior of f(x) as x approaches some number c.
Read as “the limit of the function f as x approaches c.” The symbol lim means we are taking the limit of something. The expression to the right is the expression we are taking the limit of. The expression x→c below lim means we are taking the limit of f as values of x approach c.
As values increase towards a, this is called approaching from the left (notated x→a−).
As values decrease towards a, this is called approaching from the right (notated x→a+).
When a limit doesn’t approach the same value from both sides, we say that the limit doesn’t exist.
Formal definition of a limit
means that to establish a specific limit, a number ϵ>0 is chosen first to establish a desired degree of proximity to L, then a number δ>0 is found that determines how close x must be to c to ensure that f(x) is within ϵ units of L.
General rule for the derivative of exponential function: write it down x derivative of the exponent x natural log of the base
Derivatives of logarithmic functions
General rule for the derivative of natural log functions: 1 over the object x derivative of the object
dxdlogax use the base-changing formula first
4 Applications of Derivatives
4.5 L’Hôpital’s Rule
L’Hôpital’s rule is used to evaluate indeterminate forms. L’Hôpital’s rule states if the x→climg(x)f(x) yields one of the indeterminate forms of the type 00 of ∞∞, then you can evaluate the limit by replacing f with f′ and g with g′. Because the rule can be applied repeatedly, you can keep applying it until you get something other than an indeterminate form.
A function G is an antiderivative of f on an interval I if
for all x in I.
If G is an antiderivative of f, then so is G+C for any constant C given that the derivative of any constant is 0.