Limits

The Idea of a Limit #

What the function closes in on, arrival or not.

A limit is the value a function approaches, whether or not it ever gets there

Halve the gap each time. The values approach 2 without ever reaching it.

A limit at x = 2 asks what happens near 2, never what happens exactly at 2.

This function is undefined at x = 2, yet it approaches 4. The hole does not stop the limit.

One Sided Limits #

The left and the right may disagree.

Approaching from the left and from the right can give two different answers

Approaching from the left, the values settle on −1. From the right, they settle on +1.

The two sides give different answers, so there is no single value to approach.

A limit only exists when the two sides agree. That is the whole condition.

The Limit Laws #

Limits pass through sums, products and quotients.

Limits pass straight through sums, products and quotients

The limit of a sum is the sum of the limits, so you may take each limit separately.

Products behave the same way: the limit of a product is the product of the limits.

Quotients carry one condition: the limit of the denominator must not be zero.

Indeterminate Forms #

Zero over zero decides nothing on its own.

Zero over zero decides nothing, so the expression has to be rewritten first

Substitute x = 2 into $\left(x^2-4\right)/\left(x-2\right)$(x² − 4) / (x − 2) and you get 0 over 0 — a signal, not an answer.

Factor the numerator and the (x − 2) cancels, because x is near 2 and never equal to 2.

Now substitute x = 2 into x + 2. The limit is 4.

Limits at Infinity #

Far out, only the highest power matters.

For very large x only the highest power matters, so the rest can be ignored

This curve climbs toward 2 and then flattens off, never quite reaching it.

Divide numerator and denominator by the highest power of x present.

Every $\frac{1}{x}$1/x term shrinks to 0, leaving the ratio of the leading coefficients.

The Squeeze Theorem #

Trapped between two functions sharing a limit.

A function trapped between two others that share a limit must share that limit

Suppose one function sits above another, with a third caught between them.

If the upper and lower functions approach the same value, the middle one has nowhere else to go.

That is how a function that oscillates endlessly near 0 can still have a single limit there.

The Limit of sin x over x #

The ratio that unlocks the trig derivatives.

Near 0, sin x over x approaches 1 — the limit the trigonometric derivatives rest on

Substituting x = 0 gives 0 over 0, which is not an answer: the limit must be found another way.

As x approaches 0 the curve approaches height 1, but the point at x = 0 is missing.

Near 0 the ratio is trapped between cos x and 1, and both of those approach 1.

In radians sin x is very close to x near 0, which is what makes $\frac{d}{dx}\sin x=\cos x$d/dx sin x = cos x work.

Small-Angle Approximations #

Near zero, the curves flatten into polynomials.

Near 0, sine, cosine and tangent can each be replaced by a simple polynomial

Near 0, sin x is close to x. Drawn together, the curve and the line almost coincide.

Cosine falls away from 1 like the parabola $1-\frac{x^2}{2}$1 − x²/2: flat at its maximum, then falling.

Tangent follows: a numerator near x over a denominator near 1 is nearly x itself.

At x = 0.1 the approximation 0.1 matches sin x to three decimal places.