[Calculus Series Part 2] Week 1: Limits of Exponential, Log, and Trig Functions

In the first week of calculus we step into the world of transcendental functions: functions such as exponentials, logarithms, and trigonometric functions that go beyond ordinary polynomial expressions. Beyond polynomials, we now meet exponential, logarithmic, and trigonometric functions in full. The goal this week is not to memorize a bag of formulas, but to follow one connected storyline: why $e$ is special, why the derivative of $\ln x$ becomes $1/x$, and why the limit $\sin x/x$ unlocks trig differentiation.

The series context also matters. In Part 1 we reviewed limits as the language of calculus. In this post we use that same limit idea to build the derivative formulas for new families of functions, and next week we will use those results to differentiate sine and cosine directly.

1. Lesson 1: Limits and derivatives of exponentials and logarithms

1.1 Why the natural base e?

One of the most important constants in calculus is the natural base $e$. The driving question is simple: can we choose a base $a$ so that the derivative of $a^x$ equals $a^x$ itself?

Start from the derivative definition:

The important point is that the factor

depends only on the base $a$, not on $x$. So the derivative of every exponential has the form "the same function times a constant." For example,

The unique base that makes that constant exactly $1$ is $e$, so

In other words, $e^x$ is the unique exponential whose rate of change at each point equals its value at that point, which is why it appears everywhere in calculus.

When a phenomenon grows proportionally to its current size—population growth, continuous compounding, radioactive decay, solutions of differential equations—it inevitably looks like $e^x$ or a scaled version of it. The constant $e$ is therefore not a convenience—it acts like a fundamental unit of continuous change.

1.2 Definitions of e and logarithmic differentiation

There are many equivalent ways to define $e$, but courses usually present two at once.

This is the limit of continuous compounding. If you compound interest more and more frequently-semiannually, quarterly, monthly, and ultimately infinitely often-you naturally arrive at that limit. This limit converges to approximately $2.71828$.

Another view comes from the differential equation $y' = y$ with $y(0)=1$; the base that satisfies this condition becomes $e$. These two viewpoints describe the same number. The compounding formula gives an intuitive limit model, while the differential-equation view explains why $e^x$ is the natural exponential for calculus.

Now connect this to logarithms. Since $\ln x$ is the inverse of $e^x$, let

Then $e^y = x$. Differentiate both sides with respect to $x$:

Because $e^y = x$, this becomes

So

Here $\ln x$ is special because it is the inverse of $e^x$. If $e^x$ is the most natural exponential, then $\ln x$ is the most natural logarithm. Common logarithms (base 10) or binary logarithms (base 2) are useful, but derivatives are simplest with $\ln x$.

Keep these correspondences in mind:

• Exponentials model repeated multiplicative growth, such as compounding by the same percentage.
• Logarithms turn products into sums, which is why they simplify multiplication-based relationships.
• Derivatives look simplest with $e^x$ and $\ln x$, because no extra conversion factor is needed.

1.3 Quick computation tour

See how the formulas work immediately:

The first combines an exponential derivative with the chain rule; the second pairs logarithmic differentiation with the chain rule; the third reminds us that every non-natural base injects a $\ln a$ factor below.

n_values = np.array([1, 10, 100, 1000, 10000, 100000])
e_approx = (1 + 1 / n_values) ** n_values

print("n\t(1+1/n)^n")
print("-" * 25)
for n, e_val in zip(n_values, e_approx):
    print(f"{n}\t{e_val:.10f}")

print(f"\nActual e: {np.e:.10f}")

1.4 Trig addition formulas and a key limit

Addition formulas for sine and cosine are the doors to later derivative formulas. When we differentiate $\sin x$ from first principles, $\sin(x+h)$ appears, so we need a way to expand that expression.

Visualize them as rotations on the unit circle: the formulas are not arbitrary, they express how coordinates change under rotation. We need this structure so we can expand $x+h$ in terms of $x$ and $h$ and plug it into limit definitions.

From these formulas we derive the most important limit in this unit:

Seeing how $\sin x / x$ approaches 1 as $x \to 0$ (in radians) is essential because this limit becomes the starting point of trig differentiation.

1.5 Proving the fundamental limit

The crucial part is not the answer itself but the reasoning for

We need this limit to justify trig derivatives from the definition. The standard proof uses the unit circle plus the squeeze theorem. The core geometric idea is simple: on the unit circle, the straight chord is shorter than the arc, and the arc is shorter than the tangent segment built outside the circle.

For $0<x<\frac{\pi}{2}$, compare three regions with central angle $x$:

• An inscribed triangle with area $\frac{1}{2}\sin x$
• A sector with area $\frac{1}{2}x$ because sector area on the unit circle is $\frac{1}{2}r^2\theta = \frac{1}{2}x$
• A larger triangle formed with the tangent line, with area $\frac{1}{2}\tan x$

Therefore,

and multiplying by 2 gives

Using $\tan x = \frac{\sin x}{\cos x}$,

Since $\sin x > 0$ on that interval, divide all terms by $\sin x$:

All three quantities are positive, so taking reciprocals reverses the inequalities and gives

As $x \to 0$, $\cos x \to 1$, and the squeeze theorem yields

We performed the proof for $x>0$, but $\sin(-x)=-\sin x$ shows that the same value holds from the left, so the two-sided limit exists.

This result gives us two other limits immediately:

and

so

These two limits reappear directly in next week's derivations of $(\sin x)'=\cos x$ and $(\cos x)'=-\sin x$. Remember one warning: these limits equal the stated values only in radians. If you measure $x$ in degrees, the numbers differ. In fact,

not $1$. Radians keep arc length and angle linked, which is why trig derivatives stay clean.

It is also useful to keep the following together:

For the second limit we already used

and the first limit is the auxiliary piece we need when differentiating cosine.

2. Seeing this week as one stream

The week’s ideas are tightly linked:

1. Exploring exponential derivatives reveals the special base $e$.
2. Its inverse $\ln x$ provides the simplest logarithmic derivative.
3. In trigonometry, the addition formulas plus key limits prepare us to differentiate sine and cosine.

So this week is less about "introducing new functions" and more about gathering the raw materials needed to build derivative formulas.

3. Weekly checkpoints

Focus on connecting $e$, exponential/log derivatives, trig addition formulas, and the limit $\lim_{x \to 0} \frac{\sin x}{x}$.

Checkpoint 1

Find the derivative of $f(x) = e^{2x} \ln x$.

Checkpoint 2

Compute $\lim_{x \to 0} \frac{\tan x}{x}$.

Checkpoint 3

Use the key limit to find $\lim_{x \to 0} \frac{\sin(3x)}{x}$.

Checkpoint 4

Differentiate $f(x) = e^x + \ln x - \cos x$.

Checkpoint 5

Explain in your own words why $\lim_{x \to 0} \frac{\sin x}{x} = 1$ launches trig differentiation.

4. Frequent pitfalls

• Forgetting the $\ln a$ factor and assuming $\frac{d}{dx} a^x = a^x$.
• Dropping the $\ln a$ in the denominator when differentiating $\log_a x$.
• Applying $\lim_{x \to 0} \frac{\sin x}{x} = 1$ to degree measure instead of radians.
• Ignoring that $\ln x$ is defined only for $x>0$ and pretending it differentiates over all reals.

5. Next week

Next week dives into trig differentiation and the chain rule properly. We will connect the addition formulas and limits from this week to explicit derivative formulas.