Math, asked by yasenroyce, 7 months ago

lim x➡️1 (xsin{x})/x-1​

Answers

Answered by Anonymous
1

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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