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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Limits
NDA 2018 · Mathematics · Limits
What is _ h arrow 0 2 x+3 h - 2 x 2 h equal to?
What is
lim
h
→
0
2
x
+
3
h
−
2
x
2
h
\lim _{h \rightarrow 0} \frac{\sqrt{2 x+3 h}-\sqrt{2 x}}{2 h}
lim
h
→
0
2
h
2
x
+
3
h
−
2
x
equal to?
Options
A
1
2
2
x
\frac{1}{2 \sqrt{2 x}}
2
2
x
1
B
3
2
x
\frac{3}{\sqrt{2 \mathrm{x}}}
2
x
3
C
3
2
2
x
\frac{3}{2 \sqrt{2 x}}
2
2
x
3
D
3
4
2
x
\frac{3}{4 \sqrt{2 x}}
4
2
x
3
Correct answer
D.
3
4
2
x
\frac{3}{4 \sqrt{2 x}}
4
2
x
3
Step-by-step solution
Let
L
=
lim
h
→
0
2
x
+
3
h
−
2
x
2
h
L=\lim _{h \rightarrow 0} \frac{\sqrt{2 x+3 h}-\sqrt{2 x}}{2 h}
L
=
lim
h
→
0
2
h
2
x
+
3
h
−
2
x
Rationalise the numerator, we get:
L
=
lim
h
→
0
(
2
x
+
3
h
−
2
x
2
h
×
2
x
+
3
h
+
2
x
2
x
+
3
h
+
2
x
)
L=\lim _{h \rightarrow 0}\left(\frac{\sqrt{2 x+3 h}-\sqrt{2 x}}{2 h} \times \frac{\sqrt{2 x+3 h}+\sqrt{2 x}}{\sqrt{2 x+3 h}+\sqrt{2 x}}\right)
L
=
lim
h
→
0
(
2
h
2
x
+
3
h
−
2
x
×
2
x
+
3
h
+
2
x
2
x
+
3
h
+
2
x
)
=
lim
h
→
0
2
x
+
3
h
−
2
x
2
h
(
2
x
+
3
h
+
2
x
)
=
3
2
(
2
x
+
0
+
2
x
)
=
3
4
2
x
\begin{aligned} & =\lim _{h \rightarrow 0} \frac{2 x+3 h-2 x}{2 h(\sqrt{2 x+3 h}+\sqrt{2 x})} \\ & =\frac{3}{2(\sqrt{2 x+0}+\sqrt{2 x})}=\frac{3}{4 \sqrt{2 x}}\end{aligned}
=
h
→
0
lim
2
h
(
2
x
+
3
h
+
2
x
)
2
x
+
3
h
−
2
x
=
2
(
2
x
+
0
+
2
x
)
3
=
4
2
x
3
$$
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