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CBSEClass–11Mathematics

NCERTSolutions

Chapter-13LimitsandDerivative

Exercise13.1

EvaluatethefollowinglimitsinExercises1to22.

1.

Ans. 3+3=6

2.

Ans.

3.

Ans.

4.

Ans.

5.

Ans.

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6.

Ans. isoftheform

Put nowas

=

since

=5

7.

Ans.

=

=

8.

Ans. isoftheform

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=

=

=

9.

Ans.

10.

Ans. isoftheform

=

=

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=

= =1+1=2

11.

Ans.

=

= =1

12.

Ans. =

=

= =

13.

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Ans.

=

=

= and

=

14.

Ans.

=

= since

=

15.

Ans.

Put nowas

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=

= =

=

=

16.

Ans. =

17.

Ans. isoftheform

=

=

=

18.

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Ans.

=

=

=

=

19.

Ans. =

= = =0

20.

Ans.

Dividingnumeratoranddenominatorby

=

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=

=

=

21.

Ans.Given:

= =

= =

= =0

22.

Ans.Given:

Put nowas

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=

= =

=

=

=

23.Find and where

Ans.Given:

Forx>0Righthandlimit=

Forx<0Lefthandlimit=

As ,wehave

Forx>1Righthandlimit=

Forx<1Lefthandlimit=

As ,wehave

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24.Find where

Ans.Given:

Forx>1Righthandlimit=

Forx<1Lefthandlimit=

As ,wehave doesnotexist

25.Evaluate where

Ans.Given:

Wehave whenxispositive

Forx>0Righthandlimit=

Wehave whenxisnegative

Forx<0Lefthandlimit=

As ,wehave doesnotexist

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26.Find where

Ans.Given:

Wehave whenxispositive

Forx>0Righthandlimit=

Wehave whenxisnegative

Forx<0Lefthandlimit=

As ,wehave doesnotexist

27.Find where

Ans.Given:

L.H.L.

Putting as

=

= =0

R.H.L.

Putting as

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=

= =0

Here,L.H.L.=R.H.L.

Therefore,thislimitexistsat and =0

28. Suppose and if what are possible

valuesof and ?

Ans.Given: and

and

and

and

Onsolvingtheseequation,weget and

29. Let be fixed real numbers and define a function

What is ? For some

compute

Ans.Given:

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Now

=

= =0

=0

Also

=

=

30.If forwhatvaluesof does exists?

Ans.Given:

Consider

When

L.H.L.= =

AlsoR.H.L.= =

Here,L.H.L. R.H.L.

Therefore,thislimitdoesnotexistat

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Whena>0,

L.H.L.=

AlsoR.H.L.=

Here,L.H.L.=R.H.L.

Therefore,thislimitexistatx=awhena>0

Whena>0,

L.H.L.=

AlsoR.H.L.=

Here,L.H.L.=R.H.L.

Therefore,thislimitexistatx=awhena<0

existsforall

31.Ifthefunction satisfies evaluate

Ans.

Since ,weget

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32. If forwhat integer and doesboth

and exist?

Ans.Lefthandlimit=

Righthandlimit=

Thus existsonlyifm=n

Lefthandlimit=

Righthandlimit=

As weget existforanyintegralvalueofmand

n