Learning Objectives
- Understand the concept of limits and evaluate them
- Learn standard limit results and L'Hopital-like approaches
- Study derivatives from first principles and standard rules
- Apply product rule, quotient rule for differentiation
Key Concepts
Limits
Definition: limx→a f(x) = L means f(x) approaches L as x approaches a. Left-hand limit: limx→a- f(x). Right-hand limit: limx→a+ f(x). Limit exists if LHL = RHL.
Algebra of limits: lim[f(x) ± g(x)] = lim f(x) ± lim g(x). lim[f(x) · g(x)] = lim f(x) · lim g(x). lim[f(x)/g(x)] = lim f(x) / lim g(x) (if denominator ≠ 0).
Standard Limits
limx→a (xn - an)/(x - a) = nan-1 (for any rational n).
limx→0 sin x / x = 1. limx→0 (1 - cos x)/x2 = 1/2. limx→0 tan x / x = 1.
limx→0 (ex - 1)/x = 1. limx→0 (ax - 1)/x = ln a. limx→0 log(1 + x)/x = 1.
limx→∞ (1 + 1/x)x = e. limx→0 (1 + x)1/x = e.
Methods to Evaluate Limits
Direct substitution: Try substituting the value first. Factorization: Factor and cancel common terms (for 0/0 form). Rationalization: Multiply by conjugate for expressions with square roots. Standard forms: Use standard limit results. Sandwich theorem: If g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x) = L, then lim f(x) = L.
Derivatives
First Principles: f'(x) = limh→0 [f(x+h) - f(x)] / h. This gives the instantaneous rate of change.
Standard derivatives: d/dx(xn) = nxn-1. d/dx(sin x) = cos x. d/dx(cos x) = -sin x. d/dx(tan x) = sec2x. d/dx(cot x) = -csc2x. d/dx(sec x) = sec x tan x. d/dx(csc x) = -csc x cot x. d/dx(ex) = ex. d/dx(ln x) = 1/x. d/dx(constant) = 0.
Rules: Sum/Difference: (f ± g)' = f' ± g'. Product rule: (fg)' = f'g + fg'. Quotient rule: (f/g)' = (f'g - fg')/g2. Scalar multiple: (cf)' = cf'.
Summary
Limits describe the behavior of functions as variables approach specific values. Standard limit results for algebraic, trigonometric, and exponential functions are fundamental. Derivatives measure instantaneous rate of change and are computed using first principles or differentiation rules.
Important Terms
- Limit: Value that a function approaches as input approaches a point
- Derivative: Instantaneous rate of change; slope of tangent line
- First Principles: f'(x) = limh→0 [f(x+h) - f(x)]/h
- Product rule: (fg)' = f'g + fg'
- Quotient rule: (f/g)' = (f'g - fg')/g2
- Sandwich theorem: Squeeze theorem for evaluating limits
Quick Revision
- lim(sinx/x) = 1 as x→0
- lim(xn-an)/(x-a) = nan-1 as x→a
- lim(1+1/x)x = e as x→∞
- d/dx(xn) = nxn-1; d/dx(sinx) = cosx; d/dx(cosx) = -sinx
- Product rule: (uv)' = u'v + uv'
- Quotient rule: (u/v)' = (u'v - uv')/v2
- Derivative from first principles: limh→0 [f(x+h)-f(x)]/h