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Question
If `f(x) = {{:(sin[x]/[x]",", [x] ≠ 0),(0",", [x] = 0):}`, where [.] denotes the greatest integer function, then `lim_(x -> 0) f(x)` is equal to ______.
Options
1
0
– 1
None of these
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Solution
If `f(x) = {{:((sin[x])/([x])",", [x] ≠ 0),(0",", [x] = 0):}`, where [.] denotes the greatest integer function, then `lim_(x -> 0) f(x)` is equal to none of these.
Explanation:
Given, `f(x) = {{:((sin[x])/([x])",", [x] ≠ 0),(0",", [x] = 0):}`
L.H.L = `lim_(x -> 0) (sin[x])/([x])`
= `lim_(h -> 0) (sin[0 - h])/([0 - h])`
= `lim_(h -> 0) (-sin[-h])/([-h])` = – 1
R.H.L = `lim_(x -> 0^+) (sin[x])/([x])`
= `lim_(h -> 0) (sin[0 + h])/([ 0 + h])`
= `lim_(h -> 0) (sin[h])/([h])` = 1
L.H.L ≠ R.H.L
So, the limit does not exist.
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