Answer:
[tex]\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)} = 1[/tex]
General Formulas and Concepts:
Pre-Calculus
Calculus
Limits
Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)}[/tex]
Step 2: Evaluate
- Rewrite limit: [tex]\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)} = \lim_{x \to 0} \frac{\sin (ax)}{\frac{\sin (ax)}{\cos (bx)}}[/tex]
- Simplify: [tex]\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)} = \lim_{x \to 0} \cos (ax)[/tex]
- Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)} = cos(0)[/tex]
- Simplify: [tex]\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)} = 1[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits
