Question 16 of 70intermediate🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→0ax+xcos⁡xbsin⁡x\lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}

Correct Answer

a+1b\frac{a + 1}{b}

Exercise: EXERCISE 12.1 | Q: 18 | (Page Page 22)
For More Understanding

Explanation

This question tests the application of standard limits and algebraic simplification. The key standard limit used is lim(x→0) sin x / x = 1, which implies lim(x→0) x / sin x = 1. Students need to factor out x from the numerator and separate the limit into a product of simpler limits.

Solution Steps

  1. Step 1: Simplify the expression by factoring x from the numerator: lim(x→0) [x(a + cos x)] / [b sin x]

  2. Step 2: Rewrite as: lim(x→0) [(a + cos x) / b] × [x / sin x]

  3. Step 3: Apply standard limits: As x → 0, cos x → cos 0 = 1, and lim(x→0) x/sin x = 1/lim(x→0) sin x/x = 1/1 = 1

  4. Step 4: Calculate the final limit: lim(x→0) (ax + x cos x)/(b sin x) = [(a + 1)/b] × 1 = (a + 1)/b

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