Special Trig Limits | Calculus I Mastery

There are two special trig limits that you absolutely must memorize. They appear everywhere in calculus - especially when finding derivatives of sin and cos.

The Two Essential Trig Limits

Manipulation Required

Medium

Find lim[x->0] sin(3x)/x

1

Notice the mismatch

We have sin(3x) but dividing by x, not 3x

2

Multiply by 3/3

sin(3x)/x * (3/3) = 3 * sin(3x)/(3x)

3

Apply the formula

lim[x->0] 3 * sin(3x)/(3x) = 3 * 1 = 3

Answer: 3

💡 The coefficient pops out! In general: lim[x->0] sin(ax)/x = a

Practice Problems

1. Find lim[x->0] sin(4x)/x

1

4

1/4

0

Rewrite as 4 * sin(4x)/(4x) = 4 * 1 = 4

2. Find lim[x->0] sin(x)/sin(2x)

2

1/2

1

0

= [sin(x)/x] / [2*sin(2x)/(2x)] = 1 / (2*1) = 1/2

Key Takeaways

MEMORIZE: lim[x->0] sin(x)/x = 1
MEMORIZE: lim[x->0] (1-cos(x))/x = 0
For sin(ax)/x, the coefficient 'a' pops out: answer is a
Always use radians in calculus!