Intuition The ONE core ideaBoth famous limits ask the same tiny question: "when an input shrinks toward zero (or a count grows toward infinity), what fixed number does a certain ratio or product settle on?" The whole topic is just learning to read that settling behaviour — so first we must be fluent in every symbol used to write it down: the angle, the radian, the sine, the ratio, the limit arrow, and the exponent.
Below is every piece of notation the parent note leans on, ordered so each one is built from the one before it. Nothing is used before it is drawn.
→ x → 0 x approaches 0 not mean "x 0 x 0 n → ∞ n 1 , 2 , 3 , …
Definition Two-sided vs one-sided — approach0 both directions
Zero has two neighbourhoods: numbers just above it (+ 0.1 , + 0.01 , … x → 0 + below it (− 0.1 , − 0.01 , … x → 0 − x → 0 same number from both sides . If the two sides disagreed, the two-sided limit would not exist.
Look at the figure. On the top line the burnt-orange dots creep toward 0 right (x → 0 + and the plum dots creep toward 0 left (x → 0 − teal dots march off toward larger and larger n → motion , not a destination you reach.
Intuition Why two sides matter forx s i n x x s i n x even function: swapping x − x sin ( − x ) = − sin x − x − s i n x = x s i n x x → 0 + identically for x → 0 − n → ∞ n
Intuition Why we need the arrow at allIf we could just plug x = 0 x s i n x 0 0 sneak up on 0 →
lim lim x → 0 ( some expression in x ) (some expression) settles on as x 0 from either side ". The little $x\to 0$ under the word lim tells you which way the input is moving — and with no + − both ways at once .
Worked example Reading it aloudx → 0 lim x sin x = 1 "as x x s i n x 1 We will justify the 1
This is the single most important symbol in the whole chapter , and the parent topic uses it in every line.
Definition Radian — the honest way to measure an angleDraw a circle of radius 1 unit circle , so its radius r = 1 ( 1 , 0 ) along the curved edge . The radian measure of an angle is simply how far you walked along that edge . Walk a distance of 1 1 2 π 2 π
Intuition WHAT to observe in the figureThe orange thick arc is the distance you walked; the teal radius points to where you stopped. The two labels agree: arc length = 1 and angle = 1 are literally the same number on the picture. That equality is what radians buy you — and it is only true in radians. In degrees, walking a distance of 1 ≈ 57.3° x s i n x → 1 x Radian Measure for the full story.
sin x x down to the horizontal axis . The length of that vertical drop is sin x sin x height of the point above the horizontal axis.
Intuition WHAT to observe in the figureFor a tiny angle x
the orange arc (length = x
the teal chord (the straight shortcut across),
the plum vertical (length = sin x
Follow the plum segment: it is the height, and it is barely shorter than the orange arc beside it. When x = x = sin x same length , so their ratio x s i n x 1 Squeeze (Sandwich) Theorem .
cos x tan x
cos x horizontal position of the point on the unit circle (how far right it is).tan x = cos x sin x edge of the angle , extended, hits the vertical tangent line at ( 1 , 0 )
Intuition Why the proof needstan x small triangle (height sin x big triangle (height tan x cos x < 1 tan x = c o s x s i n x bigger than sin x Derivative of sin and cos is built straight out of this same limit.
b a b a "how many times does b a A limit of a ratio, like x s i n x as both top and bottom shrink, do they stay in step? If top and bottom shrink at the same speed, the ratio settles on 1 0
e e e ≈ 2.71828 not assume it in advance; we define it as exactly the value the compounding expression settles on:
e := lim n → ∞ ( 1 + n 1 ) n . π e The Number e and ln .
Definition "Indeterminate" = the notation alone can't decide
0 0 0 cannot read off the answer — it could be 0 1 7 ∞ speeds . x s i n x x → 0 both sides, which agree because the function is even, §1) is a 0 0 1 1 ∞ 1 ∞ n → ∞ 1 + n 1 1 n 1 + tiny compounds over infinitely many multiplications. ( 1 + n 1 ) n e
Intuition WHAT to observe in the figureTwo panels. Left: the curve x s i n x teal line at 1 as x → 0 both sides — the naive guess 0 Right: the dots ( 1 + n 1 ) n orange line at e ≈ 2.718 — the naive guess 1 why these limits are worth a whole topic. More cases live in Indeterminate Forms .
Definition The exponent in( 1 + n 1 ) n ( 1 + n 1 ) n multiply the number ( 1 + n 1 ) n Two things fight: the base 1 + n 1 1 n = 1 n e
Intuition The bank picture (whye n n 1 n → ∞ e ≈ 2.718 The Number e and ln .
ln e ( ) ln ln ( e a ) = a ln ( A b ) = b ln A drags an exponent down into an ordinary product. That is exactly how the parent note attacks ( 1 + n 1 ) n ln n 0 0 L'Hôpital's Rule , and the exact-polynomial view is Taylor Series .
Approach arrow x to 0 both sides
Radian measure r equals 1
Number e defined as a limit
Indeterminate 1 to infinity
Read "x → 0 "x 0 both sides but never lands on it."
What does the plain (unsigned) lim x → 0 The same settling value from the right (0 + 0 −
Why is the two-sided limit of x s i n x The function is even, so the left and right behaviour are identical.
Define a radian using the unit circle The distance walked along the curved edge of a radius-1
Why must the angle be in radians for these limits? Only in radians does arc length equal the angle, and sector area = 2 1 r 2 x 2 1 x r = 1
What is sin x The vertical height of the unit-circle point above the horizontal axis.
Why does the proof also need tan x tan x = c o s x s i n x > sin x
How is the number e e := lim n → ∞ ( 1 + n 1 ) n ≈ 2.718
What does an indeterminate 0 0 How fast the top shrinks compared to the bottom.
Why isn't 1 ∞ 1 The base is 1 + tiny
What does ln A b Turns it into b ln A
