Reformulate the function being evaluated to avoid catastrophic cancellation.

Question

In all cases, assume x is near 0.

a) (sqrt(4+x) - 2) / x

b) cuberoot(1+x) - 1

c) (1-cos x) / x^2

d) sin (a+x) - sin a

Answer 1

I am not sure what is meant by "catastrophic cancellation".

In b) and d) the expressions are continuous at 0 with limit 0 in both cases.

In a) I would use l'Hospital's rule to compute the limit at 0, which is 1/4; you could also expand the radical using the binomial theorem which will lead to the same limit.

In d) you can use the Taylor's series for cos to show that the limit is 1/2