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Evaluate the limit, $\lim _{x \rightarrow 1^{+}}\left[(x-1) \tan \left(\frac{\p...

May 15, 2024

Solution by Steps

step 1

We need to evaluate the limit

\lim _{x \rightarrow 1^{+}} \left[(x-1) \tan \left(\frac{\pi}{2} x\right)\right]

step 2

x

approaches

1

from the right,

x-1

approaches

0

from the right

step 3

We know that

\tan \left(\frac{\pi}{2} x\right)

approaches

\tan \left(\frac{\pi}{2}\right)

, which is undefined because

\tan \left(\frac{\pi}{2}\right)

tends to

+\infty

x

approaches

1

from the right

step 4

Therefore, we need to analyze the behavior of the product

(x-1) \tan \left(\frac{\pi}{2} x\right)

x

approaches

1

from the right

step 5

Let

t = x - 1

. As

x \rightarrow 1^{+}

t \rightarrow 0^{+}

. Thus, the limit becomes

\lim_{t \rightarrow 0^{+}} t \tan \left(\frac{\pi}{2} (t + 1)\right)

step 6

Simplify the argument of the tangent function:

\frac{\pi}{2} (t + 1) = \frac{\pi}{2} t + \frac{\pi}{2}

step 7

Therefore, the limit is

\lim_{t \rightarrow 0^{+}} t \tan \left(\frac{\pi}{2} t + \frac{\pi}{2}\right)

step 8

Using the trigonometric identity

\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

, we get

\tan \left(\frac{\pi}{2} t + \frac{\pi}{2}\right) = -\cot \left(\frac{\pi}{2} t\right)

step 9

t \rightarrow 0^{+}

\cot \left(\frac{\pi}{2} t\right) \rightarrow \frac{2}{\pi t}

step 10

Therefore,

\tan \left(\frac{\pi}{2} t + \frac{\pi}{2}\right) \approx -\frac{2}{\pi t}

step 11

Substituting this back into the limit, we get

\lim_{t \rightarrow 0^{+}} t \left(-\frac{2}{\pi t}\right) = \lim_{t \rightarrow 0^{+}} -\frac{2}{\pi} = -\frac{2}{\pi}

Answer

-\frac{2}{\pi}

Key Concept

Limit of a product involving a tangent function approaching an undefined value

Explanation

The key to solving this limit is recognizing the behavior of the tangent function as it approaches an undefined value and using trigonometric identities to simplify the expression.