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# Find the limit or show that it does not exist. $\displaystyle \lim_{x \to 0^+} \tan^{-1}(\ln x)$

## $=-\frac{\pi}{2}$

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this problem Number forty of the Stuart can't close a tradition section two point six I'm the limit or showed that it does not exist. Limited's experts zero from the right with the function tangent inverse of Alan of X. Now we use a property of limits where if you have a function within a function, this can also be represented as he outer function or the limit now inside of the outer function applied to the inter function. So he righted a such that we can solve this limit by solving love it on the inside and then applying it to be outer function afterward. So let's recall the grass of the function Ellen and interest Tangent. Yellen function is of this form and as it approaches here from the right, uh, as you can see it, a purchase Negative. Infinity, for this limit we know is perching negative infinity. So Denver just towards negative infinity and the universe tension function Looks like this with a horizontal Ask himto at positive, however too. And a negative carver too. So for the inverse tangent function, if we had an input, it's argument is coin towards negative infinity, then it will be approaching negative power over too. So since this inner limit is approaching it infinity and native infinity for tension inverse purchase proper too. The answer to your original limit must be, uh, negative power, too. That is your final answer.

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