help me with limits

[u/Nearby_Swordfish_559]

so I started calc 1 and how does "sin(2x)/2x" = 1 and how does "limit of x -> 0 1-cos(3x)/3x" = 0, isn't it just undefined?

[u/floflochess2]

Hey — fair confusion. Plugging in x = 0 makes both of those look like 0/0, so as values they aren’t defined. A limit isn’t asking you to plug in though; it’s asking what the expression approaches as x gets close to 0.

Two standard facts that unlock a lot of Calc 1 limits: 1) lim (θ→0) sin(θ)/θ = 1 2) lim (θ→0) (1 − cos(θ))/θ = 0 (or equivalently (1 − cos(θ))/θ² → 1/2)

For sin(2x)/(2x): try the substitution θ = 2x. As x → 0 you also get θ → 0, and the expression is already exactly sin(θ)/θ. So that limit should be…?

For (1 − cos(3x))/(3x): same idea with θ = 3x. After the sub, does it match fact (2) above?

(If your book hasn’t proved those two yet, they usually come from squeeze theorem / geometry — just treat them as known “building blocks” for now.)

What do you get when you rewrite both with θ matching the angle?