April 8th, 2009 | 
Author: Another tricky case where we cannot determine a limit is when a function oscillates infinitely fast as it approaches a point. We call such oscillations indeterminate oscillations. Consider the function
as x approaches zero from the right. The reciprocal of x is not continuous at zero because division by zero is not defined. To the right of zero, f(x) is continuous but
can’t be evaluated.
Example 1. Determine the limit
Solution 1. As x approaches zero from the right then the inverse of x approaches infinity. This limit is the same as writing
I have made the substitution u=1/x. We know that the sine of infinity does not exist because the function is always oscillating there. Therefore the original limit does not exist either.
../images/S1OX.pdf
We see that the function $\sin (x^{-1})$ does not converge as $x$ approaches zero from the right
Similarly a random continuous function has no limit at infinity because it doesn’t converge to any value. This is not because of indeterminate oscillations, but rather a random walk.
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