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Joint Entrance Examination
Mathematics
Limits, Continuity and Differentiability
Question 262
Question 262 of 401
Medium
MCQ
If function
f
(
x
)
f(x)
f
(
x
)
is differentiable at
x
=
a
x=a
x
=
a
, then
lim
x
→
a
x
2
f
(
a
)
−
a
2
f
(
x
)
x
−
a
\displaystyle\lim_{x\to a}\dfrac{x^2f(a)-a^2f(x)}{x-a}
x
→
a
lim
x
−
a
x
2
f
(
a
)
−
a
2
f
(
x
)
is:
A
−
a
2
f
′
(
a
)
-a^2f'(a)
−
a
2
f
′
(
a
)
B
a
f
(
a
)
−
a
2
f
′
(
a
)
af(a)-a^2f'(a)
a
f
(
a
)
−
a
2
f
′
(
a
)
C
2
a
f
(
a
)
−
a
2
f
′
(
a
)
2af(a)-a^2f'(a)
2
a
f
(
a
)
−
a
2
f
′
(
a
)
D
2
a
f
(
a
)
+
a
2
f
′
(
a
)
2af(a)+a^2f'(a)
2
a
f
(
a
)
+
a
2
f
′
(
a
)
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Official solution
By L'Hopital's rule (form
0
/
0
0/0
0/0
):
lim
x
→
a
2
x
f
(
a
)
−
a
2
f
′
(
x
)
1
=
2
a
f
(
a
)
−
a
2
f
′
(
a
)
\displaystyle\lim_{x\to a}\dfrac{2xf(a)-a^2f'(x)}{1}=2af(a)-a^2f'(a)
x
→
a
lim
1
2
x
f
(
a
)
−
a
2
f
′
(
x
)
=
2
a
f
(
a
)
−
a
2
f
′
(
a
)
.
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