Calculus Derivatives and Limits Math Sheet

Limits Math Help

Definition of Limit

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.

$definition of a limit$

Right Hand Limit

The following expression states that as x approaches the value c and x > c the function approaches the value L.

$right hand limit definition$

Left Hand Limit

The following expression states that as x approaches the value c and x < c the function approaches the value L.

$left hand limit definition$

Limit at Infinity

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.

$limit at negative infinity$

Properties of Limits

Given the following conditions:

$conditions for limit properties$

The following properties exist:

$limit property with constant$

$limit property with the sum of two functions$

$limit property with the multiplcation of two functions$

$limit propety with the division of two functions$

$limit property with a function raised to a power$

Limit Evaluation at +-Infinity

$limit of e raised to the x at infinity$

$limit of the natural log at infinity$

$limit of a constant over x raised to a constant$

$limit of a constant over x raised to a constant when x raised r is real$

$limit of x raised to a constant for even r$

$limit of x raised to a constant for odd r$

Limit Evaluation Methods

Continuous Functions

If f(x) is continuous at a then:

$limit of a continuous function$

Continuous Functions and Compositions

If f(x) is continuous at b:

$limit of a the composition of continous functions$

Factor and Cancel

$limit evaluation method using factoring$

L'Hopital's Rule

$limit evaluation method using L'Hopital's rule$

Derivatives Math Help

$derivative of the sum of two functions$

$derivative of a constant$

$derivative of the cosine function$

$derivative of the tangent function$

$derivative of the secant function$

$derivative of the cosecant function$

$derivative of the cotangent function$

$derivative of the inverse sine function$

$derivative of the inverse cosine function$

$derivative of the inverse tangent function$

$derivative of a constant raised to variable$

$derivative of e raised to the power of x$

$derivative of the natural log function$

$derivative of the natural log absolute value function$

$derivative of the log function$

Chain Rule Examples

These are some examples of common derivatives that require the chain rule.

$chain rule example with function raised to power$

$chain rule example with e raised to a function$

$chain rule example of the natural log of function$

$chain rule example of the sin of a function$

$chain rule example of the cosine of a function$

$chain rule example of the tangent of a function$

$chain rule example of the secant of a function$

$chain rule example of the inverse tangent of a function$