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How do you evaluate #[ ( 1 + ( r/x ) ) ^ x ]# as x approaches infinity?

Answer 1

Emma Aldridge

#lim_(x->oo)(1+r/x)^x= e^r#

Write #f(x)# as:

#(1+r/x)^x = e^(xln(1+r/x))=e^(r ln(1+r/x)/(r/x))#

Now we have:

#lim_(x->oo) ln(1+r/x)/(r/x)= lim_(y->0) ln(1+y)/y#

To evaluate this last limit we note that:

#lim_(y->0) ln(1+y)/y=lim_(y->0) (ln(1+y)-ln1)/(y-0)#

is by definition the value for #y=0# of the derivative of #ln(1+y)#, so that:

#lim_(y->0) ln(1+y)/y = [1/(1+y]]_(y=0) = 1#

And thus:

#lim_(x->oo)(1+r/x)^x= e^r#

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Answer 2

Samantha Adkison

The expression [ ( 1 + ( r/x ) ) ^ x ] can be evaluated as x approaches infinity using the concept of the limit. The limit of this expression as x approaches infinity is equal to the mathematical constant e raised to the power of r. Therefore, the evaluation of [ ( 1 + ( r/x ) ) ^ x ] as x approaches infinity is e^r.

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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.