For which real values alpha does lim {x -> 0+} x^alpha sin(1/x) exist?

It is easy to show using the epsilon - delta definition below that this limit exists for all real alpha >= 1. In fact the limit is zero in this case. The case alpha equals zero is also quite simple and the limit does not exist. Consider the two sequences

a_n = 2/((4n + 1)pi) and b_n = 2/((4n + 3)pi)

These go to zero through positive values as n --> infinity and yet f(a_n) = 1, however f(b_n) = -1, so this limit cannot exist. I AM INTERESTED IN THE CASE alpha < 0 in which case the limit diverges, but this must be shown rigorously...Thanks!

Definition [Right Hand Limit] Let I be a nonempty interval with a as its left endpoint. A function f:I --> R is said to converge to L as x approaches a from the right if for every epsilon > 0 there is a delta > 0 such that

a < x < a + delta implies |f(x) - L| < epsilon

Solution Preview

The only comment that I can give here is that we know:

If lim[f(x)]=L and lim[g(x)]=M then lim[f(x)g(x)]=LM ; (x->a)

Now, we see that:

Because sin is a bounded function and no matter what x is -1<=sin(x)<=1, then lim(sin(1/x)) ; x->0 is for sure bounded, ...

Solution Summary

A limit is shown to diverge using a rigorous proof.

Limits of Trig Functions. ... 2) The limit of f(x) = (sin x)/x as X approaches 0 is 1. ... b) Sketch the graph of the cosine function h(x) = cos x. What is the slope of ...

... 3. Solve x²-25<0. 4. What is the limit as x approaches -3 for the function f(x)=x³. ... 4. What is the limit as x approaches -3 for the function f(x)=x³. ...

... distribution and the limiting extreme-value distribution: Consider a random sample of size n from a distribution with CDF (cumulative distribution function) F(x ...

... of n, for f (n) (0) where f is each of the functions (1) sin(x ... (Q.1) It is said that a function f : R ... R if: a) f is continuous at the point a b) the limit f ( x ...

... the sensitivity ranges for the objective function coefficients and constraint quantity values using the graphical solution of the model. mossaic tiles ltd. ...

... and stereochemistry are going to put limits on the ... physical and chemical laws relate functions to structures ... the context of form and function that originally ...

... The limit for epsilon to zero exists and is given by: ... Two functions are examined ... Let's denote the action of a distribution T on a test function u as <T,u>. Then ...

... In other words, there are functions for which you can use ... In any case, we will first prove that this function is once ... at x = 0 by looking at the limit of the ...

... Find (a) an upper limit and (b) a lower limit for the zeros of the function P(x ... Find the values of the six trigonometric functions of 0 if 0 is an angle in ...