Find the following limit
We are asked to find: [ \lim_{x \to 0} \frac{1 - \cos(x^2)}{x^2 \sin(x^2)} ] First, recall the standard small-angle approximations for ____ ______ ___ ____ ______ ___ _____ ____ ________.
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Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?
Verify the second mean value theorem for the function and
in the interval
.
Show that the equation has a real root other than
Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.
a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Using the principle of mathematical induction, show that