Question

Find the limit of sin(ax)/tan(bx) as x approaches 0.

Answer 1

`\displaystyle \lim_(x \to 0) (\sin (ax))/(\tan (bx)) = 1`

General Formulas and Concepts:

Pre-Calculus

• Unit Circle

Calculus

Limits

Limit Rule [Variable Direct Substitution]:
`\displaystyle \lim_(x \to c) x = c`

Explanation:

Step 1: Define

Identify

`\displaystyle \lim_(x \to 0) (\sin (ax))/(\tan (bx))`

Step 2: Evaluate

1. Rewrite limit:
   `\displaystyle \lim_(x \to 0) (\sin (ax))/(\tan (bx)) = \lim_(x \to 0) (\sin (ax))/((\sin (ax))/(\cos (bx)))`
2. Simplify:
   `\displaystyle \lim_(x \to 0) (\sin (ax))/(\tan (bx)) = \lim_(x \to 0) \cos (ax)`
3. Limit Rule [Variable Direct Substitution]:
   `\displaystyle \lim_(x \to 0) (\sin (ax))/(\tan (bx)) = cos(0)`
4. Simplify:
   `\displaystyle \lim_(x \to 0) (\sin (ax))/(\tan (bx)) = 1`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits