CALCULUS · BEFORE DIFFERENTIATION

Limits: approaching & growth

Before we study infinitely small changes, we need to describe what a function approaches. A limit studies nearby inputs, not just the value at the target itself. Start with two moving points.

01What do the left and right sides approach?

Play or scrub the approach. The two sample inputs never equal a. Switch cases to compare agreement, a hole, a jump, oscillation and unbounded growth.

f(x)=x^2

● From the left● From the rightArrow at edge = point continues beyond the view

Animation progress0%

$x\to 1,\quad x\ne 1$

The limit asks which value f(x) approaches as x gets arbitrarily close to a, without requiring x=a. A left limit uses x<a; a right limit uses x>a. + and − in the limit notation mean sides, not the sign of the answer.

x_L=1-h,\quad x_R=1+h

h=1.2

Side	x	f(x)
Left	-0.2	0.04
Right	2.2	4.84

\lim_{x\to1^-}f(x)=1

\lim_{x\to1^+}f(x)=1

\lim_{x\to1}x^2=1=f(1)

Both sides approach 1. Polynomials are continuous, so this limit equals f(1).

DNE means “does not exist.” A finite two-sided limit exists exactly when both one-sided limits exist and equal the same finite number. A finite animation illustrates the behavior; the formulas explain why it persists arbitrarily close to a.

02Which term eventually dominates?

The lower-degree terms may still grow, but they shrink relative to the leading term. Even a large coefficient such as 400 can delay dominance without changing the eventual limit.

x	$2x^3$	$-2x^2$	1	$\frac{\|-2x^2\|}{\|2x^3\|}$
10	2000	-200	1	0.1
100	2.000e+6	-20000	1	0.01
1000	2.000e+9	-2.000e+6	1	0.001
10000	2.000e+12	-2.000e+8	1	0.0001
1.000e+6	2.000e+18	-2.000e+12	1	1.000e-6

The quadratic term does not tend to zero: its magnitude grows. Its ratio to the cubic term tends to zero. That is the precise meaning of “negligible” here.

e^x

x^2+1

x

\sqrt{x-1}

\ln x

Right edge of x-axis6

All curves use x>0, but √(x−1) only exists for x≥1 in the real numbers. On a finite interval a larger slope does not establish a higher asymptotic growth order.

For example, √(x−1) becomes extremely steep near x=1, yet it eventually grows much more slowly than x. Compare ratios as x→∞ instead.

\ln x\ll\cdots\ll x^{1/3}\ll x^{1/2}\ll x\ll x^2\ll x^3\ll\cdots\ll e^x

Here f ≪ g means f/g→0 as x→∞. Each exponent is a fixed positive number. This is an eventual growth hierarchy, not a pointwise inequality for every positive x. The two sides have a useful inverse-function pattern: exponential ↔ logarithm and powers ↔ roots, with suitable domains.

03Watch the growth gap open up.

Keep every original term. The cubic terms grow faster than the quadratic terms, just as in the growth-rate comparison above. Increase x and compare the actual values in the numerator and denominator.

R(x)=\frac{2x^3-2x^2+1}{4x^3+400x^2-1000}

x $10$

Full expression R(x)0.0418837

Cubic terms alone: 2x³ / 4x³0.5

One shared linear scale · bar length = absolute valueFull width =

40,000

NUMERATOR

2x^3

2,000

-2x^2

-200

1

1

DENOMINATOR

4x^3

4,000

400x^2

40,000

-1000

-1,000

The ruler zooms out as the numbers grow; it is shared by all six bars. Read the actual values on the right: the quadratic terms are growing too. Their bars fall behind because the cubic terms grow faster. A tiny dot marks a nonzero term below one pixel; minus signs remain in the values.

Animation progress0%

At x=10, the factor 400 makes the quadratic term in the denominator much larger than its cubic term. The limit describes what happens eventually, not what happens at every x.

\text{large }x:\quad\frac{2x^3-2x^2+1}{4x^3+400x^2-1000}\approx\frac{2x^3}{4x^3}=\frac12

\boxed{\lim_{x\to\infty}\frac{2x^3-2x^2+1}{4x^3+400x^2-1000}=\frac12}

Both leading terms are cubic, so they grow at the same order; their coefficient ratio is 2:4. The lower-growth terms become negligible by comparison. The approximation sign matters: at a finite x, the full expression has not become exactly 1/2.