gemini pro
Here's how to find the limits of x*ln(1+1/x) as x approaches 0 and positive infinity:
Case 1: x approaches 0
We cannot directly substitute x = 0 into the expression, as it results in an indeterminate form. Let's use L'Hôpital's rule:
Rewrite the expression:
x*ln(1+1/x) = ln(1+1/x) / (1/x)
Now the form is suitable for L'Hôpital's rule (both numerator and denominator approach 0 as x approaches 0).
Apply L'Hôpital's Rule:
lim (x->0) [ln(1+1/x) / (1/x)] = lim (x->0) [d/dx(ln(1+1/x)) / d/dx(1/x)]
= lim (x->0) [-1/(x+x^2) / (-1/x^2)]
= lim (x->0) [x / (1+x)] = 0
Therefore, the limit of x*ln(1+1/x) as x approaches 0 is 0.
Case 2: x approaches positive infinity
Rewrite the expression:
x*ln(1+1/x) = ln(1 + 1/x) / (1/x)
Again, this is an indeterminate form (both numerator and denominator approach 0 as x approaches infinity).
Apply L'Hôpital's Rule:
lim (x-> ∞) [ln(1 + 1/x) / (1/x)] = lim (x-> ∞) [d/dx(ln(1 + 1/x)) / d/dx(1/x)]
= lim (x-> ∞) [-1/(x+x^2) / (-1/x^2)]
= lim (x-> ∞) [x / (x+1)]
= 1 (dividing both numerator and denominator by x)
Therefore, the limit of x*ln(1+1/x) as x approaches positive infinity is 1.
