How do you find where #f(x)=1/ln(x)# is continuous and differentiable on which interval(s)?

Answer 1

It's continuous and differentiable on the union #(0,1)\cup (1,infty)#. That is, for all #x>0# except #x=1#.

The function #ln(x)# is continuous and differentiable for all #x>0#. Therefore, #1/(ln(x))# will be continuous and differentiable for all such values of #x# as well, except for those values of #x# where #ln(x)=0#. The only such value of #x# where the logarithm is zero is #x=1#.
Therefore, #1/(ln(x))# is continuous and differentiable on the union #(0,1)\cup (1,infty)#. That is, for all #x>0# except #x=1#.

Moreover, if you are interested, the Quotient Rule gives

#d/dx(1/(ln(x)))=(ln(x) * 0-1 * 1/x)/(ln(x))^2=-1/(x (ln(x))^2)# for #x>0# and #x!=1#.
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Answer 2

The function f(x) = 1/ln(x) is continuous and differentiable on the interval (1, ∞).

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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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