Answer to Question #253963 in Calculus for Shambo

Question

Use the precise def i nition of a limit (à Ž µ à ¢ ˆ ’ à Ž ´ method) to show that lim xà ¢ † ’a x2à ¢ ˆ ’ 2ax = à ¢ ˆ ’a2. [7]

Answer 1

limit f(x) = L as x-->a

For every E>0 there exists Delta_E such that

L-E < f(x) < L+E when a-Delta_E < x < a+Delta_E

Working backwards:

If

16-E < 10-2x < 16+E

Then

$6-E < -2x < 6+E\\ (1/2)E - 3 > x > (-1/2)E - 3\\ -3 + (1/2)E > x > -3 - (1/2)E\\ -3 - (1/2)E < x < -3 + (1/2)E\\ -3 - E/2 < x < -3 + E/2\\$

For E>0, Let Delta_E = E/2

When -3-Delta_E < x < -3+Delta_E

$-3-E/2 < x < -3 + E/2\\ -6 - E < 2x < -6 + E\\ E+6 > -2x > 6-E\\ E+16 > 10-2x > 16-E\\ E+16 > f(x) > 16-E\\$