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

Answer 1

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.