Question

(a) What is wrong with the following equation? x2 + x − 42 x − 6 = x + 7 (x − 6)(x + 7) ≠ x2 + x − 42 The left-hand side is not defined for x = 0, but the right-hand side is. The left-hand side is not defined for x = 6, but the right-hand side is. None of these — the equation is correct. (b) In view of part (a), explain why the equation lim x → 6 x2 + x − 42 x − 6 = lim x → 6 (x + 7) is correct. Since x2 + x − 42 x − 6 and x + 7 are both continuous, the equation follows. Since the equation holds for all x ≠ 6, it follows that both sides of the equation approach the same limit as x → 6. This equation follows from the fact that the equation in part (a) is correct. None of these — the equation is not correct.

Answer 1

Answer:

Attached please find answer

Step-by-step explanation:

= (x² + x - 42) / (x - 6)

= [x² + (7x - 6x) - 42] / (x - 6)

= [x² + 7x - 6x - 42] / (x - 6)

= [(x² + 7x) - (6x + 42)] / (x - 6)

= [x(x + 7) - 6(x + 7)] / (x - 6)

= [(x + 7)(x - 6)] / (x - 6)

= x + 7 → when x approches 6

= 6 + 7

= 13

Other method:

Lim   (x² + x - 42) / (x - 6)

x→ 6

When x → 6, there is an indeterminate expression because: 6 - 6 = 0

Do you know the l'Hôpital's rule, when x → a:

Lim [ f(x) / g(x) ] = Lim [ f'(x) / g'(x) ]

f(x) = x² + x - 42

f'(x) = 2x + 1

g(x) = x - 6

g'(x) = 1

= f'(x) / g'(x)

= (2x + 1)/1

= 2x + 1 → when x approches 6

= 12 + 1

= 13