# Indeterminate forms of Limits

Examples with detailed solutions and exercises that solves limits questions related to indeterminate forms such as : ∞ / ∞, 0^{ 0}, ∞^{ 0}, 1^{ ∞}, ∞^{ o} and ∞ - ∞.

## TheoremAsecond version of L'Hopital's rule allows us to replace the limit problem ∞ / ∞ with another simpler problem to solve.
If lim f(x) = ∞ and lim g(x) = ∞ and if lim [ f'(x) / g'(x) ] has a finite value L , or is of the form + ∞ or - ∞, then lim [ f(x) / g(x) ] = lim [ f'(x) / g'(x) ] lim stands for lim _{x→a}, lim_{x→a+}, lim_{x→a-}, lim_{x→ + ∞} or lim_{x→ - ∞}.
## Example 1Find the limit lim_{x→∞} ln x / x
Solution to Example 1:Since lim _{x→∞} ln x = ∞
and lim _{x→∞} x = ∞The above L'Hopital's rule can be used to evaluate the given limit question lim _{x→∞} ln x / x = lim_{x→∞} [ d ( ln x ) / dx ] / [ d ( x ) / dx ]
= lim _{x→∞} [ 1 / x ] / 1 = 0
## Example 2Find lim_{x→∞} x e^{ -x}Solution to Example 2:Note that lim _{x→∞} x = + ∞
and lim _{x→∞} e^{ - x} = 0
This is the indeterminate form ∞ ^{ . }0. The idea is to convert it into to the indeterminate form ∞ / ∞lim _{x→∞} x e^{ -x}
= lim_{x→∞} x / e^{ x}Now apply the above L'Hopital's theorem lim _{x→+∞} x / e^{ x} = lim_{x→+∞} 1 / e^{ x} = 0
## Example 3Find lim_{x→∞} ( 1 + 1/x)^{ x}
## Example 4Find the limit lim_{x→0+} (1 / x - 1 / sin x)
Solution to Example 4:Note that lim _{x→0+} 1 / x = +∞
and lim _{x→0+} 1 / sin x = +∞
This limit has the indeterminate form ∞ - ∞ and has to be converted to another form by combining 1 / x - 1 / sin x lim _{x→0+} (1 / x - 1 / sin x) = lim_{x→0+} [ (sin x - x) / (x sin x) ]
We now have the indeterminate form 0 / 0 and we can use the L'Hopital's theorem. lim _{x→0+} [ (sin x - x) / (x sin x) ]
= lim _{x→0+} [ (cos x - 1) / (sin x + x cos x) ]
We have the indeterminate form 0 / 0 and use the L'Hopital's theorem again. lim _{x→0+} [ (sin x - x) / (x sin x) ]
= lim _{x→0+} [ (cos x - 1) / (sin x + x cos x) ]
= lim _{x→0+} [ (-sin x) / (cos x + cos x - x sin x) ] = 0 / 2 = 0
## Example 5Find the limit lim_{x→ 0+} x^{ x}Solution to Example 5:We have the indeterminate form 0 ^{0}. Let y = x^{ x} and ln y = ln (x^{ x}) = x ln x. Let us now find the limit of ln ylim _{x→0+} ln y
= lim _{x→0+} x ln x
The above limit has the indeterminate form 0 ^{.} ∞. We have convert it as followslim _{x→0+} x ln x
= lim _{x→0+} ln x / (1 / x)
It now has the indeterminate form ∞ / ∞ and we can use the L'Hopital's theorem lim _{x→0+} ln x / (1 / x)
= lim _{x→0+} (1 / x) / (- 1 / x^{ 2})
= lim _{x→0+} -x = 0
The limit of ln y = 0 and the limit of y = x ^{x} is equal to e ^{ 0} = 1
## ExercisesFind the limits1. lim _{x→∞} (ln x)^{ 1/x}2. lim _{x→∞} [ ln x - ln (1 + x) ]
3. lim _{x→∞} x / e^{ x}4. lim _{x→0+} x ^{ sin x}## Solutions to Above Exercises1. 12. 0 3. 0 4. 1 ## More Links on LimitsCalculus Tutorials and ProblemsLimits of Absolute Value Functions Questions |