Question 9 of 70beginner🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→0ax+bcx+1\lim_{x \to 0} \frac{ax + b}{cx + 1}

Correct Answer

b

Exercise: EXERCISE 12.1 | Q: 9 | (Page Page 21)
For More Understanding

Explanation

This is a straightforward limit evaluation using direct substitution. Since the function f(x) = (ax + b)/(cx + 1) is a rational function and the denominator cx + 1 equals 1 (not zero) when x = 0, we can directly substitute x = 0. The numerator becomes a(0) + b = b, and the denominator becomes c(0) + 1 = 1. Therefore, the limit equals b/1 = b.

Solution Steps

  1. Step 1: Check if direct substitution is possible. At x = 0, denominator = c(0) + 1 = 1 ≠\neq 0, so direct substitution is valid.

  2. Step 2: Substitute x = 0 in the numerator: ax + b = a(0) + b = b.

  3. Step 3: Substitute x = 0 in the denominator: cx + 1 = c(0) + 1 = 1.

  4. Step 4: The limit = b/1 = b.

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