Dear student fx=sinxxx≠00x=0Consider LHLlimx→0-fx=limx→0-sinxxLet x=0-hwhere h→0⇒limh→0sin0-h0-h⇒limh→0sin-h-h⇒limh→0-sinh-h⇒limh→0sinhh=1RHLlimx→0+fx=limx→0+sinxxLet x=0+h, where h→0=limh→0sin0+hh=limh→0sinhh=1 Regards

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f (x) = \{\begin{cases} \frac{\sin [x]}{[x]} & [x] \neq 0 \\ 0 & [x] = 0 \end{cases} Consider LHL \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{-}} \frac{\sin [x]}{[x]} Let x = 0 - h where h \to 0 \Rightarrow \lim_{h \to 0} (\frac{\sin [0 - h]}{[0 - h]}) \Rightarrow \lim_{h \to 0} \frac{\sin (- h)}{- h} \Rightarrow \lim_{h \to 0} \frac{- \sinh}{- h} \Rightarrow \lim_{h \to 0} \frac{\sinh}{h} = 1 RHL \lim_{x \to 0^{+}} f (x) = \lim_{x \to 0^{+}} \frac{\sin [x]}{[x]} Let x = 0 + h, where h \to 0 = \lim_{h \to 0} \frac{\sin [0 + h]}{[h]} = \lim_{h \to 0} \frac{\sinh}{h} = 1

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