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Question
Evaluate the following limits:
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
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Solution
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = lim_(x -> 3) ((x + 3)(x - 3))/(x^2(x - 3)2`
= `lim_(x -> 3) (x +3)/(x^2(x - 3))`
To find he left limit
Put x = 3 – h
Where h > 0
When x → 3
We have h → 0
`lim_(x -> 3^-) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = lim_("h" -> 0) (3 - "h" + 3)/((3 - "h")^2 (3 - "h" - 3))`
= `lim_("h" -> 0) (6 - "h")/(-"h"(3- "h")^2`
= `- lim_("h" -> 0) (6 - "h")/("h"(3 - "h")^2`
= `- (6 - 0)/(0(3 - 0)^2`
= `- 6/0`
`lim_(x -> 3^-) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = - oo`
To find he right limit
Put x = 3 + h
Where h > 0
When x → 3
We have h → 0
`lim_(x -> 3^+) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = lim_("h" -> 0) (3 + "h" + 3)/((3 + "h")^2 (3 + "h" - 3))`
= `lim_("h" -> 0) (6 + "h")/("h"(3 + "h")^2`
= `lim_("h" -> 0) (6 + "h")/("h"(3 + "h")^2`
= `(6 + 0)/(0(3 + 0)^2`
= `6/0`
`lim_(x -> 3^+) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = oo`
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