# Circular logic in evaluation of elementary limits

Our calculus book states elementary limits like $\lim_{x\to0} \frac{\sin x}{x}=1$, or $\lim_{x\to0} \frac{\ln (1+x)}{x}=1$, $\lim_{x\to0} \frac{e^x-1}{x}=1$ without proof.

At the end of the chapter of limits, it shows that these limits can be evaluated by using series expansion (which is not in our high school calculus course).

However, series expansion of a function can only be evaluated by repeatedly differentiating it.

And, to calculate derivative of $\sin x$, one must use the $\lim_{x\to0} \frac{\sin x}{x}=1$.

So this seems to end up in a circular logic. It is also same for such other limits.

I found that $\lim_{x\to0} \frac{e^x-1}{x}=1$ can be proved using binomial theorem.

How to evaluate other elementary limits without series expansion or L'Hôpital Rule?

This answer does not explain how that limit can be evaluated.

• Can you give us a proof of the assertion "..series expansion of a function can only be evaluated by repeatedly differentiating it."?
– user170039
Commented May 12, 2017 at 5:26
• I wanted to mean Taylor series or Maclaurin Series, not binomial expansion. Should I update the question that way? One can write the power series $$f(x) = a_0 + a_1(x-a) + a_2(x-a)^2 + \dots + a_n(x-a)^n + \dots$$ , and differentiate both sides to evaluate $a_i$ Commented May 12, 2017 at 5:27
• You can say that these are circular if the functions are defined via some means other than series. If these are defined via series then there is no circularity. Moreover there are many choices to define these functions and each has its own advantages and challenges. None of these is more correct than the other. So it all boils down to how you define them. Commented May 12, 2017 at 6:32
• Moreover note that the proper development of the theory of these functions (exponential, logarithmic and circular) is a non trivial task and mostly not suitable for a beginner in calculus. It is preferable to be intellectually honest and mention the properties of these functions including these limits without proof and say that proofs will be provided in advanced courses. On the hand if a textbook on introductory calculus proves these limits via L'Hospital's Rule or Taylor's series then that is an intellectual fraud. Commented May 12, 2017 at 6:49
• Do have a look at the blog posts paramanands.blogspot.com/2014/05/… and paramanands.blogspot.com/2016/03/… if you are interested in the theory of these functions. Commented May 12, 2017 at 6:53

Let $A_1$ be the area of triangle $ABC$, $A_2$ be the area of arc $ABC$, and $A_3$ be the area of ABD. Then we have:
$$A_1<A_2<A_3$$
Try and find expressions for $A_1,A_2,$ and $A_3$ and fill them into the inequality and finally use the squeeze theorem.