How do you use L'hospital's rule to find the limit #lim_(x->oo)x^(1/x)# ?

Answer 1

To find the limit lim_(x->∞)x^(1/x) using L'Hôpital's Rule, we can rewrite the expression as e^(ln(x) / x). Then, apply L'Hôpital's Rule by taking the derivative of the numerator and the derivative of the denominator separately.

lim_(x->∞) e^(ln(x) / x)
= lim_(x->∞) e^(ln(x) / x)
= e^lim_(x->∞) (ln(x) / x)

Now, apply L'Hôpital's Rule to the limit inside the exponential function:

lim_(x->∞) (ln(x) / x) = lim_(x->∞) (1/x) / 1
= lim_(x->∞) 1 / x
= 0

Therefore, the original limit is:

e^0 = 1

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Answer 2

The basic idea in using the rule of De l'Hospital to find indeterminate limits of powers #f(x)^(g(x))# is to rewrite it as #e^(g(x)\ln(f(x)))# and find the limit of the indeterminate product #g(x) ln(f(x))# rewriting the product as a quotient: #ln(f(x))/(1/g(x))# or #g(x)/(1/(ln(f(x)))#

If the power was indeterminate (#0^0# or #1^infty# or #infty^0#) then the obtained quotient is either indeterminate of the form #0/0# or #infty/infty#, so that the Rule of De l'Hospital applies to lift the indetermination.

In this example #x^(1/x)=e^(1/x lnx)# and #lim_{x\to infty} ln x/x=lim_{x to infty} (1/x)/1=0# by the Rule of de l'Hospital.

Thus#lim_{x \to \infty}x^(1/x)=e^0=1#

See this video on indeterminate powers for more:

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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