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Evaluate \mathrm{ \quad \lim _{x \rightarrow 0} \frac{x^3 e^{\frac{x^4}{4}}-\sin ^{\frac{3}{2}} x^2}{x^7}}

Option: 1

3/2


Option: 2

1/2


Option: 3

0


Option: 4

5/6


Answers (1)

best_answer

You are on the right track. By using little-o notation you may control the expansions in a better way. We have that as \mathrm{x \rightarrow 0^{+},}

\mathrm{ \frac{x^3\left(1+\frac{x^4}{4}+o\left(x^4\right)\right)-\left(x^2-\frac{x^5}{3 !}+o\left(x^6\right)\right)^{\frac{3}{2}}}{x^7}=\frac{1+\frac{x^4}{4}+o\left(x^4\right)-\left(1-\frac{x^4}{3 !}+o\left(x^4\right)\right)^{\frac{3}{2}}}{x^4} }

Now all you need is that\mathrm{ (1+t)^{3 / 2}=1+\frac{3 t}{2}+o(t)} this can be shown by verifying that,

\mathrm{ \lim _{t \rightarrow 0} \frac{(1+t)^{3 / 2}-1}{t}=\lim _{t \rightarrow 0} \frac{(1+t)^3-1}{t\left((1+t)^{3 / 2}+1\right)}=\lim _{t \rightarrow 0} \frac{3+o(1)}{\left((1+t)^{3 / 2}+1\right.}=\frac{3}{2} . }

Posted by

Deependra Verma

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