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Problem: Let $x$ tend to $0$. Determine the order of $\sqrt{\sin^2x+x^4}$ with respect to $x$.

My attempt: Let $\alpha(x)$ and $\beta(x)$ be infinitesimals as $x$ tends to $0$. Recall that if $\lim \frac{\alpha(x)}{(\beta(x))^n}=c$, where $0<|c|<+\infty$,then the function $\alpha(x)$ is an infinitesimal of the nth order as compared with $\beta(x)$ . Here $\alpha(x)=\sqrt{\sin^2x+x^4}$ and $\beta(x)=x$. Thus we compute: $$\lim\frac{\sqrt{\sin^2x+x^4}}{x^n}=\lim \frac{\sqrt{x^2+x^4}}{x^n}=\lim\frac{x(x^2/2+1)}{x^n}$$ If $n=1$ then we get $c=1$ which satisfies the condition $0<|c|<+\infty$.

Doubt:

  • Is this the correct way of solving this problem?
  • Are their better/faster methods of obtaining the order of an infinitesimal with respect to $x$ ?
  • What is the intuitive understanding of the following statement, "The order of $\sqrt{\sin^2x+x^4}$ w.r.t $x$ is $2$."?
Student
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1 Answers1

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In this limit, $\sin (x) =x+$ asymptotically smaller corrections, so that your whole expression under the radical is $x^2+$ asymptotically smaller corrections. So the whole thing is $x+$ asymptotically smaller corrections.The meaning of the order is the largest exponent $k $ such that if you divide by $x^k $ the result is still bounded as $x \to 0$. (If you measure the order with some $\beta (x)$ then you divide by $\beta (x)^k $ instead.) For larger orders the "infinitesimal" as you call it is going to zero faster.

Ian
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  • Thank you for your answer. What techniques should one use in order to determine the order? Taylor series... And also will every $\alpha(x)$ have an order, for instance does $\alpha(x)=\frac{x}{x-1}$ have an order? – Student Sep 20 '16 at 13:46
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    Taylor expansion is the most common technique; L'Hospital's rule is essentially the same thing. There are tricks you can use to ignore irrelevant terms like I did. If you include the possibility of the order being $+\infty$ or $-\infty$, then yes there will always be an order (because the order is the supremum of a well-defined set of real numbers). – Ian Sep 20 '16 at 14:01
  • @Ian -- Are there any introductory sources on the topic of "order of infinitesimals"? I'm recently began calculus/analysis and will need some good source on this sub-topic. – Fine Man Mar 02 '17 at 23:18