### Limit Function - Take the limit as x approaches ...

 Use DynamicAnalytic Geometry Material on limit Take a limit by dragging x and seeing where f(x) is going Take a limit by epsilon-delta Take a limit by examining f(x) above and below x Take a limit by the slope of the secant Take a limit at negative or positive infinity Examine the example at the bottom of this page

 Limit is a function. and Reading Limit Notation       The argument is the thing on which (or with which) the function is operated or performed. In the limit expression below, 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 a zero at - 5 because setting the numerator to zero and solving x+5 = 0, results in x = - 5. 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. a y-intercept at 5/2 because replacing x with 0 results in (0+5)/(0+2) or 5/2.         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.   Click here to see an animation on the division.

Think APPROACH to take a limit.

BY DEFINITION, if the limit is not labeled "from above" or "from below," one must approach the limit from above and from below and these values must be equal for a limit to exist.

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

See an animation or

 Do it yourself, take the limit yourself with limit.

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

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

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

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

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