"Foundations of Engineering Mathematics: First-Year Summary"

Indeteminate Forms:

To evaluate the limits of the form ∞−∞, we take L.C.M and rewrite the given expression to
0 ∞
obtain either ∨ form and then apply the L’Hospitals rule:
0 ∞

Worked Examples:

1. Evaluate 1 
lim   cot x 
x 0 x
 

Solution:
1  [ ∞−∞ form ]
lim   cot x 
x 0 x
  0 
sinx−xcosx  form 
y=lim
x→0 xsinx 0 

On Applying L'Hospital's Rule, We get
xsinx
y=lim
x→0 xcosx+ sinx


0 
 form 
y=lim
xcosx +sinx 0 
x→0 2cosx −xsin x


 y 0
1 
lim   cot x 
Thus, x 0 x
 =0

, 2. Evaluate lim  sec x  tan x 

x
2

Solution:
 1 sin x 
lim  sec x  tan x  lim       form  form]
[ ∞−∞
  cos x cos x 
x x 
2 2


y= lim
1−sinx 0 
π cosx  form 
0
x→
2



On Applying L'Hospital's Rule, We get








Thus lim  sec x  tan x  =

x
2