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Question
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
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Solution
f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
f(x) = `((x + 2)(x - 2))/((x + 2)^2 (x + 3))`
f(x) = `(x - 2)/((x +2)(x +3))`
o find the let imit of f(x) at x = – 2
Put x = – 2 – h
Where h > 0
When x → – 2
We have h → 0
`lim_(x -> - 2^-) f(x) = lim_("h" -> 0) ((-2 - "h")- 2)/((-2 "h" + 2)(- 2 - "h" + 3)`
= `lim_("h" -> 0) (-4 - "h")/((- "h")(1 - "h"))`
=`lim_("h" -> 0) 1/"h" ((4 + "h")/(1 - "h"))`
= `1/0 ((4 + 0)/(1 - 0))`
= `oo`
`lim_(x -> - 2^-) f(x) = oo`
o find the right limit of f(x) at x = – 2
Put x = – 2 + h
Where h > 0
When x → – 2
We have h → 0
`lim_(x -> - 2) f(x) = lim_("h" -> 0) ((-2 + "h") - 2)/((-2 + "h" + 2)(-2 + "h" + 3))`
= `lim_("h" -> 0) (-4 + "h")/("h"(1 + "h"))`
= `lim_("h" -> 0) 1/"h"(("h" - 4)/(1 +"h"))`
= `1/0 ((0 - 4)/(1 + 0))`
= `- oo`
`lim_(x -> - 2^-) f(x) = - oo`
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