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Question
Find the left and right limits of f(x) = tan x at x = `pi/2`
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Solution
To find the left limit of f(x) at x = `pi/2`
Put x = `pi/2 - "h"`
Whee h > 0
When `x -> pi/2`
We have h → 0
`lim_(x -> pi^-/2) (tan x) = lim_("h" -> 0) tan(pi/2 - "h")`
= `lim_("h" -> 0) cot "h"`
= cot (0)
= `oo`
`lim_(x -> pi^-/2) (tan x) = oo`
To find the right limit of f(x) at x = `pi/2`
Put x = `pi/2 + "h"`
Whee h > 0
When `x -> pi/2`
We have h → 0
`lim_(x -> pi^+/2) (tan x) = lim_("h" -> 0) tan(pi/2 + "h")`
= `lim_("h" -> 0) (- cot"h")`
= `- lim_("h" -> 0) cot "h"`
`lim_(x -> pi^+/2) (tan x)` = – cot 0
= `- oo`
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