If `f(x)={{:((sin[x])/([x])","" ""for "[x]ne0),(0","" ""for "[x]=0):}`where `[x]` denotes the greatest integer less than or equal to x. Then find `lim

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If `f(x)={{:((sin[x])/([x])","" ""for "[x]ne0),(0","" ""for "[x]=0):}`where `[x]` denotes the greatest integer less than or equal to x. Then find `lim

If `f(x)={{:((sin[x])/([x])","" ""for "[x]ne0),(0","" ""for "[x]=0):}`where `[x]` denotes the greatest integer less than or equal to x. Then find `lim_(xto0)f(x).`

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - Limit does not exist.
We have `f(x)={{:((sin[x])/([x])" ""if "x in(-oo", "0)uu[1", "oo)),(0" "" if "x in[0", "1)):}`
`:." "underset(xto0^(-))limf(x)=underset(hto0)lim(sin[-h])/( [-h])=underset(hto0)lim(sin(-1))/((-1))=sin1`
and `underset(xto0^(+))limf(x)=underset(hto0)lim0=0`
Thus `underset(xto0-)limf(x)neunderset(xto0+)limf(x)`
`:." "underset(xto0^(-))limf(x)` does not exist.