Let a₁, a₂, …, aₙ be fixed real numbers and define a function f(x) = (x - a₁)(x - a₂) … (x - aₙ). What is lim x → a 1 f(x)? For some a ≠ a₁, a₂, …, aₙ, compute lim x → a f(x).

Let a₁, a₂, …, aₙ be fixed real numbers and define a function f(x) =…

Question 27 of 70intermediate🔧 ApplyShort Answer2 marks

Let $a_1, a_2, \ldots, a_n$ be fixed real numbers and define a function $f(x) = (x - a_1)(x - a_2) \ldots (x - a_n)$ . What is $\lim_{x \to a_1} f(x)$ ? For some $a \neq a_1, a_2, \ldots, a_n$ , compute $\lim_{x \to a} f(x)$ .

Correct Answer

(i) 0

(ii) $(a - a_1)(a - a_2) \ldots (a - a_n) = f(a)$

Exercise: EXERCISE 12.1 | Q: 29 | (Page Page 22)

For More Understanding

Explanation

This question tests the understanding of limits of polynomial functions. For part (i), when x approaches a₁, the factor (x - a₁) approaches 0, making the entire product approach 0. For part (ii), when x approaches a point a that is not equal to any of the roots a₁, a₂, ..., aₙ, each factor approaches a finite non-zero value, so the limit is simply the value of the function at that point.

Solution Steps

Step 1: For $\lim_{x \to a_1} f(x)$ , observe that $f(x) = (x - a_1)(x - a_2) \ldots (x - a_n)$ . As $x \to a_1$ , the factor $(x - a_1) \to 0$ .
Step 2: All other factors $(x - a_2), (x - a_3), \ldots, (x - a_n)$ approach finite values $(a_1 - a_2), (a_1 - a_3), \ldots, (a_1 - a_n)$ respectively.
Step 3: Since one factor approaches 0, the entire product approaches 0. Thus, $\lim_{x \to a_1} f(x) = 0$ .
Step 4: For $\lim_{x \to a} f(x)$ where $a \neq a_1, a_2, \ldots, a_n$ , each factor $(x - a_i)$ approaches $(a - a_i)$ , which is non-zero.
Step 5: The limit equals the product of these values: $\lim_{x \to a} f(x) = (a - a_1)(a - a_2) \ldots (a - a_n) = f(a)$ .

Previous Next