Limits and Derivatives
1. Intuitive Idea - Getting Close
Think of your friend standing at the gate. You walk towards them. You are at 10 meters, then 5, then 1, then 0.1 meter. You never say you have reached, but everyone can see where you are heading. That heading point is the limit.
Limit = value f(x) approaches as x tends to a. Left = Right = L then limit exists. Function need not be defined at a.
• Idea: As x→a, f(x)→L
• Check: Left=Right=L
• Note: f(a) may be undefined
• Example: (x²-1)/(x-1) at 1 →2
2. Algebra of Limits
Once we know limits of simple things, we can build big ones. If lim f = L and lim g = M, then sum L+M, difference L-M, product L·M, quotient L/M if M not zero.
lim(f+g)=L+M, lim(f-g)=L-M, lim(fg)=LM, lim(f/g)=L/M M not zero
• Sum: L+M
• Diff: L-M
• Product: LM
• Quotient: L/M if M not zero
3. Limits of Polynomials and Rational
Polynomials: direct substitution. Rational 0/0: factor and cancel. Example (x²-4)/(x-2) → x+2 →4. For x→∞, degree check.
Poly direct sub. Rational 0/0 factor cancel. x→∞ deg check.
• Poly: P(a)
• 0/0: Factor (x-a) cancel
• ∞: Deg check
4. Limits of Trigonometric Functions
Most used: lim x→0 sin x / x =1 radians. So sin 5x/5x→1, sin5x/x→5. Trick sin7x/sin3x =7/3.
lim sin x/x=1, (1-cos x)/x²=1/2, tan x/x=1. sin(kx)/kx=1, sin(kx)/x=k.
• Main: sin x/x=1
• Derived: (1-cos x)/x²=1/2
• Scale: sin(kx)/x=k
5. Exponential and Log Limits
lim (e^x-1)/x=1, ln(1+x)/x=1. For small x, e^x approx 1+x, ln(1+x) approx x. (a^x-1)/x→ln a.
(e^x-1)/x=1, ln(1+x)/x=1, (a^x-1)/x=ln a, (1+1/x)^x=e.
• e: (e^x-1)/x=1
• ln: ln(1+x)/x=1
• a^x: →ln a
6. Derivative First Principle
Derivative is speedometer. f'(x)=lim h→0 [f(x+h)-f(x)]/h. Secant → tangent. x²→2x.
f'(x)=lim h→0 [f(x+h)-f(x)]/h. Secant→tangent. Velocity=ds/dt.
• Def: f'(x)=lim h→0 [f(x+h)-f(x)]/h
• Idea: Average→instantaneous
7. Algebra of Derivatives
(f+g)'=f'+g', (fg)'=f'g+fg', (f/g)'=(f'g-fg')/g².
(f±g)'=f'±g', (kf)'=kf', (fg)'=f'g+fg', (f/g)'=(f'g-fg')/g².
• Sum: f'+g'
• Product: f'g+fg'
• Quotient: (f'g-fg')/g²
8. Standard Derivatives
x power n → n x power n-1, sin→cos, cos→-sin, tan→sec², e^x→e^x, ln x→1/x.
d(x^n)=n x^(n-1), sin→cos, cos→-sin, tan→sec², e^x→e^x, ln→1/x.
• Power: n x^(n-1)
• Trig: sin→cos, cos→-sin
• Exp: e^x→e^x, ln→1/x
9. Real Life Usage
Limits: battery →100%. Derivatives: speedometer, max height, profit marginal, road steepness, stock change.
Limit battery, average→instantaneous. Derivative velocity, slope, max/min profit, growth rate.
• Limit: Charging →100%
• Derivative: Speed, growth, profit, slope
• Limit: sin x/x=1, (e^x-1)/x=1, ln(1+x)/x=1
• Derivative: f'(x)=lim h→0 [f(x+h)-f(x)]/h
• Algebra: (fg)'=f'g+fg', (f/g)'=(f'g-fg')/g²
• Standard: x^n→n x^(n-1), sin→cos
• Real: Speed, growth, profit