limit is a function

8/6/2019 Limit is a Function

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Limit is a function.

andReading Limit Notation

A function is a really dependable rule.

The argumentis the thing on which (or with which) the function is operated or performed. In the limit expressionbelow, most would say the argument is the function (x+5)/(x+2). The limiting constant, 2, is the "unstated argument."

See an animation on how to read limit notation.

Use division to transform the expression for easy graphing.

The function f(x) = (x+5)/(x+2) can be easily seen to have

y a zero at - 5because setting the numerator to zero and solving

x+5 = 0, results in x = - 5.

y a vertical asymptote at x = - 2because setting the denominator to zero and solving

x+2=0, results in x = - 2, so the function is undefined at -2.

y a y-interceptat 5/2because replacing x with 0 results in (0+5)/(0+2) or 5/2.


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More than that may not be easy to see, but, a little bit of long division makes the rational expression look more like a

function that is easy to graph. Clickhere to see an animation on the division.

Think APPROACH to take a limit.

For continuous (and some other) functions, taking a limit requires one simply to approach, get closer and closer, toevaluate the limit.

See an animation.

Look at the graph of the function then take a limit graphically. Click on the expression to view the answer.

BY DEFINITION one must approach the limit above and below and these values must be equal for a limit to be

evaluated.

Approach the limit from above.


evaluated.

Look at the graph. Approach the limit from above and check the answer by clicking on the expression.


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Approach the limit from below.


evaluated.

Look at the graph. Approach the limit from below and check the answer by clicking on the expression.

Take a limit.


evaluated.

Look at the graph. State the limit and check the answer by clicking on the expression.


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Take a limit at infinity.

Look at the graph. Take a limit at negative infinity. Check the answer.


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Calculus Derivatives and Limits Math Sheet


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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.

Right 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

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

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.

Also the limit as x approaches negative infinity, the value of c is a very large and negative number, is

expressed below.

Properties of Limits

Given the following conditions:

The following properties exist:

Limit Evaluation at +-Infinity


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Limit Evaluation Methods

Continuous Functions

If f(x) is continuous at a then:

Continuous Functions and CompositionsIf f(x) is continuous at b:

Factor and Cancel

L'Hopital's Rule

Derivatives Math Help

Definition of a Derivative

The derivative is way to define how an expressions output changes as the inputs change. Using limits the

derivative is defined as:

Mean Value Theorem

This is a method to approximate the derivative. The function must be differentiable over the interval (a,b) and

a < c < b.

Basic Properites

If there exists a derivative for f(x) and g(x), and c and n are real numbers the following are true:

Product Rule

The product rule applies when differentiable functions are multiplied.

Quotient Rule

Quotient rule applies when differentiable functions are divided.


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Power Rule

The power rule applies when a differentiable function is raised to a power.

Chain Rule

The chain rule applies when a differentiable function is applied to another differentiable function.

Common Derivatives


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Chain Rule ExamplesThese are some examples of common derivatives that require the chain rule.


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DIFFERENTIATION USING THE QUOTIENT RULE

The following problems require the use of the quotient rule. In the following discussion and

solutions the derivative of a functionh(x) will be denoted by orh'(x) . The quotientrule is a formal rule for differentiating problems where one function is divided by another. It

follows from the limit definition of derivative and is given by

.

Remember the rule in the following way. Always start with the ``bottom'' function and

end with the ``bottom'' function squared. Note that the numerator of the quotient ruleis identical to the ordinary product rule except that subtraction replaces addition. In

the list of problems which follows, most problems are average and a few aresomewhat challenging.

o PROBLEM 1 : Differentiate .ClickHERE to see a detailed solution to problem 1.



o PROBLEM 4 : Differentiate .


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ClickHERE to see a detailed solution to problem 4.

o PROBLEM 5 :Differentiate .




Some of the following problems require use of the chain rule.


o PROBLEM 9 : Consider the function .Evaluate .



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o PROBLEM 15 : Differentiate


o PROBLEM 16 : Find an equation of the line tangent to the graph

of at x=-1 .


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o PROBLEM 17 : Find an equation of the line tangent to the graph

of at .


o PROBLEM 18 : Consider the function . Solvef'(x) = 0 forx .Solve f''(x) = 0 forx .


o PROBLEM 19 : Find all points (x, y) on the graph of wheretangent lines are perpendicular to the line 8x+2y = 1 .


limit is a function

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