Math Help

Indeterminate Forms - (continuing discussion)

Type 3: (product) The limit of the function involves 0 . ∞ or ∞ . 0

Procedure: Convert to a quotient function 0/0 or ∞/∞ using division; You have

then converted the indeterminate for to a Type 1 where L’Hospital’s Rule applies.

Example: lim [x ln x] = lim [( ln x ) / 1/x ] = lim [ (1/x ) / ( ˗ 1/x²) ]

x → 0 x → 0 x → 0

lim [ ˗ x ] = 0

x → 0

Type 4: (power) The limit of the function involves 0⁰ , ∞⁰ , and 1 ^∞

Procedure: To evaluate the limit in type 4 first take the logarithm of the

function. Then you attempt to put the indeterminate form into a quotient function

(Type 1) and apply L’Hospital’s Rule.

Steps: Let y = [ f(x) ] ^g(x), take ln y = g(x) ln [ f(x)] then

evaluate lim ln y = L, then lim [ f(x) ] ^g(x)= e ^L

x → a x → a

Example: Find the limit of y = [cos x ] ^1/x as x → 0

ln y = [ (1/x) ln (cos x) ]

lim ln y = lim [ (1/x) ln (cos x) ] = lim [ ln (cos x) ] / x

x → 0 x → 0 x → 0

has the Type 1 form 0 / 0 so apply L’Hospital’s rule directly.

lim [ {- sin x / cos x} / 1 ] = lim ln y = 0

x → 0 x → 0

So y = lim [cos x ] ^1/x = e ⁰ = 1

x → 0