Question

cos(π/∞)=0
...Can anyone of you explain why cos(π/∞) is equal to 0 ? .......please give the solution also?

Answer 1

there is no such number as \(\frac{\pi}{\infty}\)

Answer 2

Like satellite said, that expression is not a number, but you probably mean to ask why
\[\lim_{x\to\infty}\cos\frac{\pi}{x}=0\]
which, by the way, is not a true statement.

Consider the fraction \(\dfrac{\pi}{x}\). As \(x\to\infty\), the denominator gets larger and larger, so \(\dfrac{\pi}{x}\to0\).

Since the cosine function is continuous, we can use the property that
\[\lim_{x\to0}\cos(f(x))=\cos\left(\lim_{x\to0}f(x)\right)\]
Since \(\dfrac{\pi}{x}\to0\), we're left with
\[\cos\left(\lim_{x\to0}\frac{\pi}{x}\right)=\cos0=1\not=0\]