Let g(x)= (2x)- (x) x(2 x-1) for all for which the denominator is non-zero.
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
Let
g(x)=x(2cosx−1)sin(2x)−sin(x)
for all x=0 for which the denominator is non-zero.
(a) Determine all real numbers in the interval (−π,π) where g is not defined. Give reasons that cite both the numerator and denominator.
(b) Evaluate x→0limg(x), providing a complete limit argument that may involve trigonometric identities and standard limits.
(c) State the type of discontinuity that occurs at x=0 and justify your classification.
(d) Define a new function h by assigning an appropriate value at x=0 so that h becomes continuous at x=0. Prove that your definition achieves continuity.
(e) Without graphing technology, describe the local behavior of g near x=0 and explain how this behavior is consistent with your work above.