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Answer:

\(\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)} = 1\)

General Formulas and Concepts:

Pre-Calculus

  • Unit Circle

Calculus

Limits

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

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \lim_{x \to 0} \frac{\sin (ax)}{\tan (bx)}\)

Step 2: Evaluate

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

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

Unit: Limits

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