Question

Consider the following statements :
Statement 1 : \(\lim\limits_{x\rightarrow1}\frac{ax^2+bx+c}{cx^2+bx+a}\) is 1 (where a + b + c ≠ 0)
Statement 2 : \(\lim\limits_{x\rightarrow-2}\frac{\frac{1}{x}+\frac{1}{2}}{x+2}\) is \(\frac{1}{4}\)

• A. Only statement 2 is true
• B. Only statement 1 is true
• C. Both statements 1 and 2 are true
• D. Both statements 1 and 2 are false

Answer 1

The Correct Option is B

Let's evaluate both statements:

Statement 1: We are given that \[ \lim_{x \to 1} \frac{ax^2 + bx + c}{cx^2 + bx + a} \] Substituting \( x = 1 \) into the expression, we get \[ \frac{a(1)^2 + b(1) + c}{c(1)^2 + b(1) + a} = \frac{a + b + c}{c + b + a} = 1 \] Thus, statement 1 is true.

Statement 2: We are given that \[ \lim_{x \to -2} \frac{1}{x + 2} \] As \( x \to -2 \), the denominator approaches 0, making the limit undefined. Therefore, statement 2 is false.

The correct answer is (B) : Only statement 1 is true