Three useful limit formulas, their meaning, and how to derive related limits

Three useful starting points

Three standard limits organize many elementary calculations:

$$\lim_{x\to0}\frac{\sin x}{x}=1,$$ $$\lim_{x\to0}\frac{e^x-1}{x}=1,$$ $$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e.$$

Angles in the first formula are in radians, and logarithms below are natural logarithms. These are a useful small collection to understand well, rather than a claim that no other facts about limits will ever be needed.

The author's original study advice was to focus on these three formulas and derive related ones instead of memorizing a long disconnected list.

Remember the first two through their graphs

Near zero, $\sin x$ behaves like $x$. The first limit says their ratio tends to one, even though both values tend to zero.

Similarly, $e^x-1$ behaves like $x$ near zero:

These pictures help interpret the formulas as local first-order behavior. A sketch is not a proof, and a proof using tangent slopes must not presuppose a derivative formula that itself depends on the limit being proved.

The third formula describes $e$ through a limiting power. Depending on the development of calculus, it may be adopted as a definition or proved equivalent to another definition of $e$.

Derivation 1: the cosine limit

For $x\ne0$ sufficiently close to zero,

$$\frac{1-\cos x}{x^2}=\frac{1-\cos^2x}{x^2(1+\cos x)}=\frac{(\sin x/x)^2}{1+\cos x}.$$

Using the sine limit and continuity of cosine gives

$$\boxed{\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12.}$$

The useful algebraic move is multiplication by the conjugate $1+\cos x$, which creates $\sin^2x$ and exposes the known limit.

Derivation 2: the logarithmic limit

We want

$$\lim_{x\to0}\frac{\ln(1+x)}x.$$

Set $t=\ln(1+x)$, so $x=e^t-1$ and $t\to0$. On the real domain $x>-1$,

$$\frac{\ln(1+x)}x=\frac{t}{e^t-1}.$$

The exponential limit therefore yields

$$\boxed{\lim_{x\to0}\frac{\ln(1+x)}x=1.}$$

The logarithmic graph is the inverse of the exponential graph, reflected across $y=x$. In the original diagram, $\log$ denotes the natural logarithm.

Derivation 3: a power with a minus sign

For $x>1$, write $u=x-1$. Then

$$\left(1-\frac1x\right)^x=\left(\frac{u}{u+1}\right)^{u+1}=\left\{\left(1+\frac1u\right)^u\right\}^{-(u+1)/u}.$$

The base tends to $e>0$ and the exponent tends to $-1$. By continuity of positive real powers,

$$\boxed{\lim_{x\to\infty}\left(1-\frac1x\right)^x=\frac1e.}$$

The restriction $x>1$ ensures that the base is positive while taking arbitrary real powers.

What counts as a proof of the starting formulas?

The sine limit has a geometric squeeze proof. The article proving the limit of sin x/x also explains how to justify the sector-area formula without relying on trigonometric derivatives.

The exponential limit expresses that the derivative of $e^x$ at zero is one. If that derivative has already been established independently, the limit follows immediately from the definition of derivative. If the derivative was proved using the same limit, citing it in the reverse direction would be circular. The logical starting point depends on the chosen definition of the exponential function.

Likewise, the limiting-power definition of $e$ and the exponential derivative are related, but passing between them requires a justified construction of real powers and continuity. They should not be treated as unrelated formulas that automatically prove one another without those steps.

Match the variable before applying a formula

For example,

$$\frac{\sin3x}{x}=3\frac{\sin3x}{3x},$$

so its limit at zero is $3$. The same principle gives

$$\lim_{x\to0}\frac{e^{cx}-1}{x}=c$$

for a fixed real constant $c$; the case $c=0$ is immediate.

Before using a standard limit, check the direction of approach, the real domain, the units of angles, and whether the numerator and denominator use the same small variable. Those checks make a small set of formulas much more useful than memorization alone.