Angles are in radians. To what number does (cos x)^(1/x²) tend as x shrinks to 0? Round to 3 decimal places.

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Cosine to a large power

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Angles are in radians. To what number does $(cos x)^{\frac{1}{x ^{2}}}$ tend as $x$ shrinks to $0$ ? Round to $3$ decimal places.

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Hint

Take the logarithm to turn the $1 \infty$ form into a ratio.

Solution

Let $L$ be the limit; then $ln L$ is the limit of $\frac{ln ( cos x )}{x ^{2}}$ .
Near $0$ , $cos x = 1 - \frac{x ^{2}}{2} + \dots$ and $ln (1 + u) = u + \dots$ , so $ln (cos x) = - \frac{x ^{2}}{2} + \dots$ .
Hence $ln L = - \frac{1}{2}$ and $L = e^{- \frac{1}{2}}$ .
Since $e^{- \frac{1}{2}} \approx 0.60653$ , the limit is $\approx 0.607$ .

Answer: $\approx 0.607$