The Limits

The Limits
A limit is the value that a function or sequence "approaches" as the input or index approaches some value.
We say that the limit of f(x) is L as x approaches a and write this as
lim?(x?a)??f(x)?=L
Properties of the limit
1. If a and c are constants then
2. lim?(x?a)?x=a
Let f_1 (x)=L_1 and f_2 (x)=L_2 then
3. lim?(x?a)?(f_1 (x)+f_2 (x))=L_1+L_2
4. lim?(x?a)?(f_1 (x)-f_2 (x))=L_1-L_2
5. lim?(x?a)?(f_1 (x).f_2 (x))=L_1.L_2
6. lim?(x?a)?((f_1 (x))/(f_2 (x) ))=L_1/L_2 ; L_2?0
7.If lim?(x?a)?g(x)=L then lim?(x?a)?f(g(x))=f(L)
Example 8: Find the following limits
1. lim?(x?3)??(?x-?3)/(x-3)?=lim?(x?3)??(?x-?3)/(?x-?3)(?x+?3) ?=lim?(x?3)??1/((?x+?3) )?=( 1 )/(2?(3 ))
2. lim?(x??)??(5x^2+3)/(7x^2+2x-5)?
Divide top and bottom by x^2, then we get
lim?(x??)??(5x^2+3)/(7x^2+2x-5)?=lim?(x??)??(5+(3?x^2 ))/(7+(2?x)-(5?x^2 ) )?=( 5 )/( 7 )
3. lim?(x??)??(2x+1)^4/(x^2+2x-3)^2 ?=(lim?(x??)??(2x+1)^2/(x^2+2x-3)? )^2=(lim?(x??)??(?4x?^2+4x+1)/(x^2+2x-3)? )^2
=(lim?(x??)??(4+(4?x)+(1?x^2 ))/(1+(2?x)-(3?x^2 ) )? )^2=4^2=16