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Mathematics
Mathematics
Problem
Evaluate the limit
L
=
lim
x
→
0
x
−
sin
x
ln
(
1
+
x
)
−
x
+
x
2
2
.
L=\lim_{x\to 0}\frac{x-\sin x}{\ln(1+x)-x+\dfrac{x^{2}}{2}}.
L
=
x
→
0
lim
ln
(
1
+
x
)
−
x
+
2
x
2
x
−
sin
x
.
(a) Show that direct substitution results in
0
/
0
0/0
0/0
. (b) Calculate the exact value of
L
L
L
and justify each step in your calculation.
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