Question

A student attempted to confirm that the function f defined by f(x)= (x^2+x-6)/x^2-7x+10 is continuous at x=2 '. In which step, if any, does an error first appear? Step 1: f(x)= (x^2+x-6)/x^2-7x+10 = (x-2)(x+3) / (x-2)(x-5) Step 2: f(x)= (x^2+x-6)/x^3-7x+10 = ((x-2)(x+3))/(x-2)(x-5) Step 3: limlimits _xto 2f(x)=limlimits _xto 2 (x+3)/x-5 = (2+3)/2-5 =- 5/3 f(2)= (2+3)/2-5 =- 5/3 Step 4: limlimits _xto 2f(x)=f(2) , so f is continuous at x=2. (A) Step 2 (B) Step 3 (C) Step 4 (D) There is no error in the confirmation.

Answer 1

The first error occurs in Step 2, where the degree of the polynomial in the denominator is incorrectly raised, which affects the continuity confirmation at x=2. The correct approach is to maintain the original degree and reassess the limit and continuity after simplifying the function correctly.

Step-by-step explanation:

The first error occurs in Step 2 of the student's attempt to confirm the continuity of the function. The error is that the denominator in the function is incorrectly rewritten as x3 instead of x2. The correct simplification should maintain the original degree of the polynomial in the denominator. Because of this mistake, the subsequent steps and the conclusion about continuity at x=2 are rendered unreliable. Proper factorization of the original equation would reveal that there is a removable discontinuity at x=2, leading to the need to reconsider Step 4. After simplifying, the limit limx→2 f(x) should be calculated again without the error.

The correction for Step 2 should be:

f(x) = (x2+x-6)/(x2-7x+10) = ((x-2) (x+3))/((x-2) (x-5)).

After removing the common factors, we have f(x) for x ≠ 2 as:

f(x) = (x+3)/(x-5).

Then continuing with the limit and evaluating the function at x=2, we find:

limx→2 f(x) = 5/ (-3) and f (2) is not defined because of the removable discontinuity.