How do you find the limit #ln(x^2+1)/x# as #x->0#?

Answer 1

Limit as x->0 of #ln(x^2+1)/x= 0#

Direct application give #0/0# So we use l'Hôpital rule #(ln(x^2+1)')/(x')=(2x)/(x^2+1)=0/1=0#
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Answer 2

l'Hopital's Rule applies.

#:. lim_(x->0)ln(2x^2 + 1)/x = 0#

If we try to evaluate at the limit we obtain: #0/0#

This means that l'Hopital's Rule applies.

To apply l'Hopital's Rule, you, compute the derivative of numerator, compute the derivative of the denominator, and then reassemble the two derivatives into a new fraction.

The derivative of the numerator:

#(d[ln(x^2+ 1)])/dx = (2x)/(x^2 + 1)#

The derivative of the denominator:

#(d[x])/dx = 1#

Here is our new expression:

#lim_(x->0) (2x)/(x^2+1)#

l'Hopital's Rule states that the limit of our new expression goes to the limit as the original expression

#:. lim_(x->0)ln(2x^2 + 1)/x = 0#
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Answer 3

To find the limit of ln(x^2+1)/x as x approaches 0, we can use L'Hôpital's Rule. Taking the derivative of the numerator and denominator separately, we get (2x)/(x^2+1) for the numerator and 1 for the denominator. Evaluating the limit of this expression as x approaches 0, we find that it equals 0. Therefore, the limit of ln(x^2+1)/x as x approaches 0 is 0.

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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.

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.

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.

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.

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