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How do you find the Limit of #ln(1+5x) - ln(3+4x)# as x approaches infinity?

Answer 1

Samantha Adkison

#log_e(5/4)#

#f(x)=log_e(1+5x) - log_e(3+4x) = log_e((1+5x)/(3+4x))#

So #lim_{x->oo}f(x) = log_e(lim_{x->oo}(1+5x)/(3+4x)) = log_e(lim_{x->oo}(1/x+5)/(3/x+4)) =log_e(5/4)#

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Answer 2

Samantha Adkison

To find the limit of ln(1+5x) - ln(3+4x) as x approaches infinity, we can use the properties of logarithms and the limit rules.

First, we can simplify the expression by using the logarithmic property ln(a) - ln(b) = ln(a/b). Applying this property, we get ln((1+5x)/(3+4x)).

Next, we can use the limit rule that states if f(x) and g(x) are functions such that lim f(x) = L and lim g(x) = M, then lim (f(x)/g(x)) = L/M, provided M is not equal to zero.

As x approaches infinity, the numerator (1+5x) and the denominator (3+4x) both tend to infinity. Therefore, we can apply the limit rule and find that the limit of ln((1+5x)/(3+4x)) as x approaches infinity is ln(5/4).

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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.