Find the following limit.
We need to compute the following limit: \[ \lim_{x \to 0} \frac{1 - \cos(x^2)}{x^2 \sin(x^2)} \] Let's break it down step by step. 1. Simplify the expression: We can _______ _________ ______ ____ ___ _________ _________ ________ ________.
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Let f : [0,1] → be a function defined by f (x) =xm (1-x)n where . m, n∈
Find the values of m and n such that the Rolle’s Theorem holds for the function f .
Using Weiestrass M-test, show that the following series converges uniformly.
Suppose that f : [0,2]→ is continuous on[0,2] and differentiable on ]0,2[ and that f(0)=0,f(1)=1,f(2)= 1.
(i) Show that there exists c1 ∈(0,1) such that .f' (c1) = 1.
(ii) Show that there exists c2 ∈(1,2) such that .f'(c2)= 0.
(iii) Show that there exists c ∈ (0,2) such that f'(c) =
Find the following limit.
Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
Also check whether the function f is derivable at x = .1
Check whether the series n is uniformly convergent or not, where . + α ∈
Check whether the series is uniformly convergent or not, where . α ∈
.