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lim_(x rarr1^(-))(x^(2)-1)^(cos((pi x)//2))=

$\lim _{x \rightarrow 1^{-}}\left(x^{2}-1\right)^{\cos ((\pi x) / 2)}=$

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Find trigonometric ratios using reference angles

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Q.

\lim _{x \rightarrow 1^{-}}\left(x^{2}-1\right)^{\cos ((\pi x) / 2)}=

Identify limit expression: Identify the limit expression: $\lim_{x \to 1^{-}}(x^2 - 1)^{\cos(\frac{\pi * x}{2})}$ .
Notice approaching zero: Notice that as $x$ approaches $1$ from the left, $x^2 - 1$ approaches $0$ .
Consider cosine function: Consider the exponent $\cos\left(\frac{\pi * x}{2}\right)$ . When $x$ is close to $1$ , $\cos\left(\frac{\pi * x}{2}\right)$ is close to $\cos\left(\frac{\pi}{2}\right)$ , which is $0$ .
Indeterminate form: The base $(x^2 - 1)$ is approaching $0$ , and the exponent is approaching $0$ . This is an indeterminate form $0^0$ .
Apply L'Hôpital's Rule: To resolve the indeterminate form, we can use L'Hôpital's Rule by taking the natural logarithm of the function and then finding the limit.
Apply natural logarithm: Apply the natural logarithm: $\ln((x^2 - 1)^{\cos(\frac{\pi * x}{2})}) = \cos(\frac{\pi * x}{2}) * \ln(x^2 - 1)$ .
Find limit of product: Now find the limit of $\cos\left(\frac{\pi \cdot x}{2}\right) \cdot \ln(x^2 - 1)$ as $x$ approaches $1$ from the left.
Limit of cosine function: The limit of $\cos\left(\frac{\pi \cdot x}{2}\right)$ as $x$ approaches $1$ from the left is $0$ .
Limit of natural logarithm: The limit of $\ln(x^2 - 1)$ as $x$ approaches $1$ from the left is $\ln(0)$ , which is undefined.

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