Topic: Continuity and Differentiability (Test 1)

Topic: Continuity and Differentiability
Q.1
limx01cosxx2 is equal to
A. 1/2
B. -1
C. 0
D. 1
Explaination / Solution:

(UsingL’hospital Rule ).

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Q.2
is equal to
A. 0
B. none of these
C. 2/3
D. -2/3
Explaination / Solution:

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Q.3
is equal to
A. -1/48
B. none of these
C. 1/24
D. 1/48
Explaination / Solution:

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Q.4
If k be an integer, then  (x –[x])
A. is equal to – 1
B. does not exists.
C. 1
D. is equal to 0
Explaination / Solution:

= k - (k - 1) = 1 for all
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Q.5

is equal to

A. none of these.
B. 0
C. 1
D. 1/2
Explaination / Solution:

( using L’Hospital Rule)

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Q.6
is equal to
A. 1/2
B. 8
C. 0
D. none of these.
Explaination / Solution:

( using L’Hospital Rule)

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Q.7

is equal to

A. 0
B. 1
C. does not exist
D. none of these
Explaination / Solution:

limx0tanxlog(1+x)=limx0tanxxxlog(1+x)=limx0tanxx.limx011xlog(1+x)=1.11=1
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Q.8
The function f (x) = 1 + | sin x l is
A. differentiable nowhere
B. differentiable everywhere.
C. continuous nowhere
D. continuous everywhere
Explaination / Solution:

is not derivable at those x for which  is continuous everywhere (being the sum of two continuous functions)
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Q.9
The function, f (x) = (x – a) sin for x a and f (a) = 0 is
A. Derivable at x = a
B. Not continuous at x = a
C. None of these
D. Continuous but not derivable at x = a
Explaination / Solution:

$\frac{\right)}{}$
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Q.10
Let f (x) = [x], then f (x) is
A. differentiable for all x∈(R−I).
B. continuous for all x∈R
C. continuous nowhere
D. differentiable for all x∈R