Evaluate the limit Lim (ln2x)x→∞
To evaluate the limit of ln(2x)/x as x approaches infinity, we can use L’Hôpital’s rule
To evaluate the limit of ln(2x)/x as x approaches infinity, we can use L’Hôpital’s rule.
L’Hôpital’s rule states that if we have a limit of the form f(x)/g(x) as x approaches infinity or negative infinity and both f(x) and g(x) approach 0 or infinity, then the limit is equal to the limit of f'(x)/g'(x).
In this case, we have f(x) = ln(2x) and g(x) = x. Both f(x) and g(x) approach infinity as x approaches infinity.
Taking the derivative of f(x) and g(x) with respect to x, we have:
f'(x) = (1/x)
g'(x) = 1
Now we can evaluate the limit of f'(x)/g'(x) as x approaches infinity:
lim (x→∞) (1/x) / 1 = 0 / 1 = 0
Therefore, lim (ln(2x)/x) as x approaches infinity is equal to 0.
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