The limit is a method of evaluating an expression as an argument approaches a value. This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity. The following expression states that as x approaches the value c the function approaches the value L.
definition of a limit
The following expression states that as x approaches the value c and x > c the function approaches the value L.
right hand limit definition
The following expression states that as x approaches the value c and x < c the function approaches the value L.
left hand limit definition
The following expression states that as x approaches infinity, the value c is a very large and positive number, the function approaches the value L.
limit at infinity
Also the limit as x approaches negative infinity, the value of c is a very large and negative number, is expressed below.
limit at negative infinity
Given the following conditions:
conditions for limit properties
The following properties exist:
limit property with constant
limit property with the sum of two functions
limit property with the multiplcation of two functions
[画像:limit propety with the division of two functions]
[画像:limit property with a function raised to a power]
limit of e raised to the x at infinity
limit of the natural log at infinity
[画像:limit of a constant over x raised to a constant]
[画像:limit of a constant over x raised to a constant when x raised r is real]
limit of x raised to a constant for even r
limit of x raised to a constant for odd r
If f(x) is continuous at a then:
limit of a continuous function
If f(x) is continuous at b:
[画像:limit of a the composition of continous functions]
[画像:limit evaluation method using factoring]
[画像:limit evaluation method using L'Hopital's rule ]
The derivative is way to define how an expressions output changes as the inputs change. Using limits the derivative is defined as:
definition of a derivative using limits
This is a method to approximate the derivative. The function must be differentiable over the interval (a,b) and a < c < b.
[画像:derivatives mean value theorem]
If there exists a derivative for f(x) and g(x), and c and n are real numbers the following are true:
derivative of a function with a constant
derivative of the sum of two functions
The product rule applies when differentiable functions are multiplied.
derivative product rule - derivative of two functions multiplied
Quotient rule applies when differentiable functions are divided.
[画像:derivative quotient rule - derivative of the division of two functions]
The power rule applies when a differentiable function is raised to a power.
[画像:derivative power rule- derivative of a function raised to the power]
The chain rule applies when a differentiable function is applied to another differentiable function.
[画像:derivative of two functions applied to one another]
[画像:derivative of the sin function]
[画像:derivative of the cosine function]
[画像:derivative of the tangent function]
[画像:derivative of the secant function]
[画像:derivative of the cosecant function]
[画像:derivative of the cotangent function]
[画像:derivative of the inverse sine function]
[画像:derivative of the inverse cosine function]
[画像:derivative of the inverse tangent function]
[画像:derivative of a constant raised to variable]
[画像:derivative of e raised to the power of x]
[画像:derivative of the natural log function]
[画像:derivative of the natural log absolute value function]
[画像:derivative of the log function]
These are some examples of common derivatives that require the chain rule.
[画像:chain rule example with function raised to power]
[画像:chain rule example with e raised to a function]
[画像:chain rule example of the natural log of function]
[画像:chain rule example of the sin of a function]
[画像:chain rule example of the cosine of a function]
[画像:chain rule example of the tangent of a function]
[画像:chain rule example of the secant of a function]
[画像:chain rule example of the inverse tangent of a function]
