A) \[a=1,\,b=2\]
B) \[a=1,b={{\left( -3 \right)}^{1/3}}\]
C) \[a=2,\text{ }b=3\frac{1}{3}\]
D) None of these
Correct Answer: B
Solution :
[b] \[\underset{h\to 0}{\mathop{\lim }}\,\frac{(1+{{a}^{3}})+8{{e}^{1/h}}}{1+(1-{{b}^{3}}){{e}^{1/h}}}=2\] \[\underset{h\to 0}{\mathop{\lim }}\,\frac{\frac{1+{{a}^{3}}}{{{e}^{1/h}}}+8}{\frac{1}{{{e}^{1/h}}}+1-{{b}^{3}}}=2\Rightarrow \frac{8}{1-{{b}^{3}}}=2\] \[\Rightarrow {{b}^{3}}=-3\,\,b=-{{3}^{1/3}}\] L.H.L. \[\underset{h\to 0}{\mathop{\lim }}\,\frac{(1+{{a}^{3}})+8{{e}^{-1/h}}}{1+(1-{{b}^{3}}){{e}^{-1/h}}}=2\Rightarrow 1+{{a}^{3}}=2\] \[\Rightarrow a=1\]
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