Limits Involving Trigonometric Functions

Overview

Lecture covers limits involving trigonometric functions near 0.
Emphasizes key limit identities, substitution techniques, and algebraic manipulation.
Demonstrates calculator checks (radian mode) and step-by-step formal work for proofs.

Numerical check: plug values (e.g., 0.1, 0.01, 0.001) in radian mode to observe approach.
Algebraic scaling: multiply numerator and denominator by constants to form sin(u)/u.
Substitution: let u = kx to convert sin(kx)/(kx) → sin u / u and change limit variable.
Separate products into limits when each limit exists.
For tangent limits, convert to sin/cos and use known sin/x identity and cos(0)=1.

Example: lim_{x→0} sin x / x
- Numerical checks: sin(0.1)/0.1 ≈ 0.998; sin(0.01)/0.01 ≈ 0.99998
- Concludes limit = 1.
Example: lim_{x→0} (1 − cos x) / x
- Numerical checks: at 0.1 ≈ 0.05, at 0.01 ≈ 0.005, values approach 0.
- Concludes limit = 0.
Example: lim_{x→0} sin(5x) / x
- Multiply numerator and denominator by 5: = lim sin(5x)/(5x) · 5
- Substitute y = 5x → lim_{y→0} sin y / y · 5 = 1·5 = 5.
Example: lim_{x→0} sin(2x) / (5x)
- Rearrange: = lim sin(2x)/(2x) · (2/5)
- Substitute y = 2x → = 1 · (2/5) = 2/5.
Example: lim_{x→0} sin(7x) / sin(3x)
- Rewrite as [sin(7x)/(7x)] · [(3x)/sin(3x)] · (7/3)
- Let y = 7x, w = 3x → factors → 1 · 1 · (7/3) = 7/3.
Example: lim_{x→0} tan x / x
- tan x = sin x / cos x → (sin x / x) · (1 / cos x)
- Take limits: 1 · (1/1) = 1.
Example: lim_{x→0} tan(4x) / (3x)
- tan(4x)/(3x) = [sin(4x)/(4x)] · [4/(3)] · [1/cos(4x)]
- Substitute y = 4x → = 1 · 4/3 · 1 = 4/3.
Example: lim_{x→0} 7(1 − cos x) / x
- Factor constant: 7 · lim (1 − cos x)/x = 7 · 0 = 0.
Example: lim_{x→0} (sin^2 x) / x
- Write sin^2 x = (sin x)(sin x) and rearrange: (sin x / x) · sin x
- Take limits: 1 · 0 = 0.
Example: lim_{x→0} sin(x^2) / x
- Multiply top and bottom by x to form sin(x^2)/x^2 · x
- Let y = x^2 → lim sin y / y · lim x = 1 · 0 = 0._

Substitution: introduce u = kx to transform limit into known form sin u / u.
Radian Mode: calculator must be in radian mode when evaluating trig limits near 0.
Direct Substitution: plug limit point into continuous factors when valid (e.g., cos 0 = 1).

Expression Pattern	Typical Manipulation	Result
sin(kx)/x	Multiply by k/k, substitute u=kx → (sin u/u)·k	k
sin(kx)/sin(mx)	Create (sin(kx)/(kx))·((mx)/sin(mx))·(k/m)	k/m
tan(kx)/(mx)	Convert tan to sin/cos, form sin(kx)/(kx), use cos(0)=1	k/m
(1−cos x)/x	Numerical check or known identity	0
sin^2 x / x	Write as (sin x / x)·sin x, use limits	0
sin(x^2)/x	Form sin(x^2)/x^2 · x, substitute y=x^2	0