Question

Q.5. Discuss the continuity of the functions:

f (x) = Cos ( ), when x o, f (o) = 0​

Answer 1

Step-by-step explanation:

ANSWER

Given:

f(x)=

⎩

⎪

⎨

⎪

⎧

xcos

x

1

;x



=0

0;x=0

To show that f(x) is continuous at x=0

or lim

x→0

f(x)=f(0)=0

Now we have to evaluate

lim

x→0

f(x)=lim

x→0

xcos

x

1

As we know that for all x∈R−{0},R is the set of real numbers.

−1≤cos

x

1

≤1

⇒−x≤xcos

x

1

≤x for x>0 ..........(1)

and x≤xcos

x

1

≤−x for x<0 ..........(2)

Now, lim

x→0

(−x)=0

and lim

x→0

x=0

From (1) and (2), we have

lim

x→0

xcos

x

1

=0 by sandwich theorem

Thus,lim

x→0

f(x)=f(0)=0 is satisfied.

Hence f(x) is continuous at x=0

Hence proved.

Answer 2

Step-by-step explanation:

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