Calculating+limits+using+algebra

=__Calculating Limits Using Algebra__=

A Limit
A limit is the range value a function gets infinitely close to as the graph gets infinitely close to a domain value.

Direct Substitution
The most basic method to calculate the limit is to plug an input into the function and obtain the output. The value will be true if there is no discontinuity at the input.

Alternatively, the denominator may not allow a certain value to be plugged in because the denominator can never equal zero. In this case, factor the zero from the numerator and denominator and proceed to re-input the input. The resulting coordinate is a hole, a removable discontinuity.

If the zero cannot be factored in the numerator, a table of values must be constructed as when approaching the number from the left or right will provide a solution equal to infinity or negative infinity. If the infinity sign is the same from the left and the right then the range approached infinity or negative infinity at the domain value.

Find the Limit as x approaches 2 of the function f(x)=x 2 +2x+1
 * Example One **

Solution: This function is continuous at x=2, so we can use direct substitution without any simplification: lim(x->2)(f(x))=f(2)=(2) 2 +2(2)+1=**9**

Find the limit as x approached 2 of the function f(x)=(x 2 -4)/(x-2)
 * Example Two **

Solution: We cannot substitute immediately because the function is undefined at x=2. However we can factor the numerator into (x-2)(x+2), and then the (x-2) term will divide out. Now we can substitute: lim(x->2) f(x)=(2)+2=**4**. Even though the function is undefined at x=2, the graph still approaches 4 as x approached 2.

Find the limit as x approaches 0 of the function f(x)=(x+3)/x 2
 * Example Three **

Solution: There is no way to divide out the x2 term from the denominator, so we will construct a table of values to approximate the limit: __x | -1 | -0.1 | -.01 | 0 | .01 | .1 | 1 |__ f(x) | 2 | 290 | 29900 | UND | 30100 | 310 | 4 | We can see as the values of x approach zero, the values of f(x) approach **infinity**.

Continue to: Estimating Limits from Graphs or Data Tables Return to: Limits of Functions