Find the limit

Question

x -> 0+ (tan9x)^x?

Answer 1

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Because direct substitution produces the indeterminate form .

To began proceed, assume that the limit exists and is equal to y.

Take natural log of each side.

Apply limit continuity

Write Indeterminate form .

Write Indeterminate form .

Apply L-Hospital Rule.

Write Indeterminate form .

Apply L-Hospital Rule.

Now, because ln y = 0, conclude that , and it follows that

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