Find the limit as x approaches infinity of (lnx)^(1/x). This unit is on L'Hopital's rule. I know that the answer is 1, I just don't know how to get there. I tried taking the ln of everything so that you have ln(the whole limit) = limx-->infinity (1/x)ln(lnx) but I don't know if that's the right step to take or not. Can someone point me in the right direction?
Nevermind you can ignore this, I figured it out. I was just forgetting to do e^answer at the end.
To find the limit as x approaches infinity of the function (lnx)^(1/x), you can indeed make use of L'Hopital's rule. However, before applying L'Hopital's rule, it's helpful to simplify the expression first.
Start by taking the natural logarithm of the given function: ln((lnx)^(1/x)). This is a useful step because logarithms often simplify exponential expressions.
Using the property of logarithms, we can bring down the exponent of 1/x in front of the logarithm, giving us (1/x)ln(lnx).
Now, let's proceed to apply L'Hopital's rule. This rule states that if we have an indeterminate form of the type 0/0 or ∞/∞, and the derivative of the numerator and denominator both exist, then the limit of the function equals the limit of the derivative of the numerator divided by the derivative of the denominator.
Taking the derivative of the numerator (ln(lnx)) and the denominator (x), we get:
[(1/lnx) * (1/x)] / 1.
This simplifies to (1/lnx) / x = 1/(x * lnx).
Next, we take the limit as x approaches infinity of 1/(x * lnx). As x goes to infinity, both x and ln(x) go to infinity, so we have an indeterminate form of ∞/∞.
To apply L'Hopital's rule again, we differentiate the numerator and the denominator separately. The derivative of x with respect to x is 1, and the derivative of ln(x) with respect to x is 1/x.
Therefore, the limit becomes (1/1)/(1 * ln(x)) = 1/(ln(x)), as x approaches infinity.
Finally, taking the limit of 1/(ln(x)) as x approaches infinity, you'll find that the logarithm grows very slowly compared to x, and eventually approaches infinity. Mathematically, this means that the limit is equal to 0.
So, the answer to the limit as x approaches infinity of (lnx)^(1/x) is indeed 1.
Keep in mind that L'Hopital's rule can be used when you have indeterminate forms, such as 0/0 or ∞/∞. However, it should be applied cautiously and when appropriate, as it may not always yield the correct result.