Evaluate the limit Lim (ln2x)x→∞
To evaluate the limit of (ln(2x))/x as x approaches infinity, we can use L’Hopital’s Rule
To evaluate the limit of (ln(2x))/x as x approaches infinity, we can use L’Hopital’s Rule.
L’Hopital’s Rule states that if we have a situation where the limit of f(x)/g(x) as x approaches infinity or negative infinity is in an indeterminate form (such as 0/0 or ∞/∞), then the limit of f(x)/g(x) as x approaches infinity or negative infinity can be evaluated by taking the derivative of the numerator and denominator, and then re-evaluating the limit.
In this case, we have the limit of (ln(2x))/x as x approaches infinity. This is in the form of 0/0, so we can use L’Hopital’s Rule.
Taking the derivative of the numerator, ln(2x), with respect to x, we get:
d/dx(ln(2x)) = 2/x.
Taking the derivative of the denominator, x, with respect to x, we get:
d/dx(x) = 1.
Now we can re-evaluate the limit:
Lim (ln(2x))/x as x→∞
= Lim (2/x)/1 as x→∞
= Lim 2/x as x→∞
As x approaches infinity, the denominator, x, becomes larger and larger. Thus, the expression (2/x) becomes closer and closer to 0. Therefore, the limit of (ln(2x))/x as x approaches infinity is 0.
Lim (ln(2x))/x as x→∞ = 0.
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