Answer to Question #315584 in Calculus for Aune

Question #315584

Evaluate the following limits, if they exist, where ⌊π‘₯βŒ‹ is the greatest integer function.



(a)lim ⌊2π‘₯βŒ‹/π‘₯


π‘₯β†’0



(b) lim π‘₯ ⌊1/π‘₯βŒ‹


π‘₯β†’0

1
Expert's answer
2022-03-25T05:17:26-0400

"a:\\\\x\\in \\left( -\\frac{1}{2},0 \\right) :\\frac{\\left[ 2x \\right]}{x}=\\frac{-1}{x}\\\\\\underset{x\\rightarrow 0-}{\\lim}\\frac{\\left[ 2x \\right]}{x}=-\\infty \\\\x\\in \\left( 0,\\frac{1}{2} \\right) :\\frac{\\left[ 2x \\right]}{x}=0\\\\\\underset{x\\rightarrow 0-}{\\lim}\\frac{\\left[ 2x \\right]}{x}=0\\\\The\\,\\,\\lim it\\,\\,doesn\u0091t\\,\\,exist\\\\b:\\\\x\\cdot \\left( \\frac{1}{x}-1 \\right) <x\\left[ \\frac{1}{x} \\right] \\leqslant x\\cdot \\frac{1}{x}\\\\\\underset{x\\rightarrow 0}{\\lim}x\\left( \\frac{1}{x}-1 \\right) =1,\\underset{x\\rightarrow 0}{\\lim}x\\cdot \\frac{1}{x}=1\\Rightarrow \\\\\\Rightarrow \\underset{x\\rightarrow 0}{\\lim}x\\left[ \\frac{1}{x} \\right] =1"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS