Introduction: From Secants to Tangents

One of the most elegant ideas in all of mathematics is this: you can find the exact slope at a single point on a curve by taking a limit. Before we get there, it helps to understand what a derivative actually represents geometrically.

Imagine drawing a straight line through two points on a curve. This is called a secant line, and its slope gives the average rate of change of the function between those two points. Now imagine sliding one point closer and closer to the other , the secant line rotates, and in the limit, it becomes the tangent line at that point. The slope of that tangent line is the instantaneous rate of change: the derivative.

This transition from average to instantaneous rate of change is the heart of differential calculus, and it is made rigorous through the concept of a limit.

Limits: The Language of Calculus

Before defining the derivative formally, we need to be comfortable with limits.

Limit: The limit of a function $f (x)$ as $x$ approaches a value $a$ is the value that $f (x)$ gets arbitrarily close to as $x$ gets arbitrarily close to $a$ (but does not necessarily equal $a$ ). Written as:
$lim_{x \to a} f (x) = L$

Limits can either converge (approach a finite value) or diverge (grow without bound or fail to settle on any value).

Example

Convergent limit: Consider
$lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$
Direct substitution gives $\frac{0}{0}$ , which is indeterminate. Factor the numerator:
$lim_{x \to 2} \frac{( x + 2 ) ( x - 2 )}{x - 2} = lim_{x \to 2} (x + 2) = 4$
The limit converges to $4$ .

Divergent limit: Consider
$lim_{x \to 0} \frac{1}{x}$
As $x \to 0^{+}$ , the function $\to + \infty$ ; as $x \to 0^{-}$ , it $\to - \infty$ . The two one-sided limits disagree, so the limit diverges (does not exist).

Exam Tip

A key algebraic technique for evaluating limits that produce $\frac{0}{0}$ is to factor and cancel the offending term. This is exactly the strategy used in first principles differentiation.

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9 more sections in this topic

Introduction: From Secants to Tangents

This transition from average to instantaneous rate of change is the heart of differential calculus, and it is made rigorous through the concept of a limit.

Limits: The Language of Calculus

Before defining the derivative formally, we need to be comfortable with limits.

Limits can either converge (approach a finite value) or diverge (grow without bound or fail to settle on any value).

Example

Exam Tip

A key algebraic technique for evaluating limits that produce $\frac{0}{0}$ is to factor and cancel the offending term. This is exactly the strategy used in first principles differentiation.

Free preview

AHL 5.12—First principles, higher derivatives

Introduction: From Secants to Tangents

Limits: The Language of Calculus

9 more sections in this topic

AHL 5.12—First principles, higher derivatives

Introduction: From Secants to Tangents

Limits: The Language of Calculus

9 more sections in this topic