NCERT Class 12-Mathematics: Chapter –5 Continuity and Differentiability Part 3

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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