lim x➡️1 (xsin{x})/x-1
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ʟᴛx->0[xsɪɴ(1/x)]
ʜᴇʀᴇ sɪɴ(1/x) ɪs ᴛʜᴇ ʙᴏᴜɴᴅᴇᴅ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ғᴜɴᴄᴛɪᴏɴ ᴀɴᴅ ғᴏʀ ᴀɴʏ ʀᴇᴀʟ ᴅᴏᴍᴀɪɴ ᴏғ x ᴛʜᴇ ʀᴀɴɢᴇ ᴏғ ᴛʜᴇ ғᴜɴᴄᴛɪᴏɴ ʟɪᴇs ɪɴ ᴛʜᴇ ɪɴᴛᴇʀᴠᴀʟ [-1,1] ᴀɴᴅ ᴛʜᴇ sɪɴᴇ ғᴜɴᴄᴛɪᴏɴ ɪs ᴄᴏɴᴛɪɴᴜᴏᴜs ᴛʜʀᴏᴜɢʜᴏᴜᴛ ᴛʜᴇ ʀᴇᴀʟ ᴅᴏᴍᴀɪɴ ᴏғ ᴛʜᴇ x.
sᴏ, ᴀs ʟɪᴍɪᴛ x ᴛᴇɴᴅs ᴛᴏ ᴢᴇʀᴏ ᴛʜᴇ ғᴜɴᴄᴛɪᴏɴ sɪɴ(1/x) ɪs ʙᴏᴜɴᴅᴇᴅ ʙᴇᴛᴡᴇᴇɴ ᴀɴʏ ʀᴀɴɢᴇ ᴏғ ᴛʜᴇ x€ [-1,1] sᴏ ᴀs x ᴛᴇɴᴅs ᴛᴏ ᴢᴇʀᴏ ᴛʜᴇ ᴠᴀʟᴜᴇ ʙᴇᴄᴏᴍᴇs ᴛʜᴇ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴢᴇʀᴏ ᴀɴᴅ sᴏᴍᴇ ғɪɴɪᴛᴇ ᴠᴀʟᴜᴇ, sᴏ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ʟɪᴍɪᴛ ɪs ᴢᴇʀᴏ.
ᴀs ᴢ= ʟᴛx->0 [xsɪɴ(1/x)]
ᴢ=0*ғɪɴɪᴛᴇ ᴠᴀʟᴜᴇ
ᴢ=0
sᴏ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ʟɪᴍɪᴛ ɪs ᴢᴇʀᴏ ᴀs x ᴛᴇɴᴅs ᴛᴏ ᴢᴇʀᴏ.
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