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NCERT Class 12- Mathematics: Chapter β 5 Continuity and Differentiability Part 3
Question 4:
Show that the function defined by
is continuous at .
Answer:
Left hand limit at is given by
Similarly
Thus, Hence is continuous at
Question 5:
Given . Find the points of discontinuity of the composite function .
Answer:
We know that is discontinuous at
Now, for ,
which is discontinuous at .
Hence, the points of discontinuity are and .
Question 6:
Let , for all . Discuss the derivability of
Answer:
We may rewrite
Now,
Now
Since the left hand derivative and right hand derivative both are equal, hence is differentiable at .
Question 7:
Differentiate
Answer:
Let . Using chain rule, we have
Question 8:
Answer:
Given . differentiating both sides w. r. t. , we have
Or
Therefore,
Question 9:
If , prove that
Answer:
Given that . Differentiating both sides , we have
Or
Which implies that