Advertisements
Advertisements
प्रश्न
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> 0^+) x^x`
उत्तर
`lim_(x -> 0^+) x^x` .......[0° indeterminate form]
Let g(x) = xx
Taking log on both sides
log g(x) = log xx
log g(x) = x log x
`lim_(x -> 0^+) log "g"(x) = lim_(x -> 0^+) x log x` .......`[0 xx oo "indeterminate form"]`
= `lim_(x -> 0^+) log x/(1/x)` ........`[oo/oo "Indeterminate form"]`
Applying L’ Hôpital’s rule,
= `lim_(x -> 0^+) (1/x)/(- 1/x^2)`
= `lim_(x -> 0^+) - x` = 0
Exponentiating, we get
`lim_(x -> ^+) "g"(x)` = e0
= 1
APPEARS IN
संबंधित प्रश्न
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> 0) (1 - cosx)/x^2`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x - oo) (2x^2 - 3)/(x^2 -5x + 3)`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> oo) x/logx`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> x/2) secx/tanx`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> oo) "e"^-x sqrt(x)`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> 0) (1/sinx - 1/x)`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> oo) (1 + 1/x)^x`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> pi/2) (sin x)^tanx`
Evaluate the following limits, if necessary use l’Hôpital Rule:
`lim_(x -> 0^+) (cos x)^(1/x^2)`
Evaluate the following limits, if necessary use l’Hôpital Rule:
If an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = `"A"_0 (1 + "r"/"n")^"nt"`. If the interest is compounded continuously, (that is as n → ∞), show that the amount after t years is A = A0ert
Choose the correct alternative:
The value of the limit `lim_(x -> 0) (cot x - 1/x)` is
