Page 2 - mathematics

The Limit of a Function Basic Properties and Rules for Limits

lim f x   L lim g x   M

x a x a

Definition lim f x   g x   lim f  x   lim  g  x   L M

limf x   L x a x a x a

x a lim c f x   c lim f x   c L

The limit of f x  as x approaches a, equal L x a x a

lim f x  g x   lim f x   lim g x   L  M

x a x a x a

lim  f x   lim f x   L , if lim  g x   0
 g x  M
x a   x a x a

x a f x L  lim  g  x

x a

 lim n  Ln

x a
f x  n  lim f x 

x a

Example Example x  9  3 2

Evaluate the following limit lim x2  9 Evaluate the following limit lim  x 
x3  27  
x3 x 0 

Solution Solution

lim x2  9  0 lim  x 9  3 2   lim x  9  32 0 * x9 3
x3  27 0  x   0 x9 3
x3 x0    x0 x  

x  3x  3
lim  3 x 2  3x  9
   x x  9  3  lim x  9  9  lim 1  1
x 3 9 6
x x 0 x x  9  3 x 0
 lim x 3
x  3
x 0
x2  3x  9
  lim  lim  x 9  32   1 2  1
x3  x   6  36
x 0  
 6
27

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