+0  
 
0
434
6
avatar

lim(x/(2-cos(x)),x->infinity

Guest Dec 26, 2015

Best Answer 

 #5
avatar+26329 
+10

The denominator varies between 1 and 3  (finite positive numbers).  The numerator tends to infinity.  Infinity divided by any finite number is still infinite.

Alan  Dec 27, 2015
Sort: 

6+0 Answers

 #1
avatar
+10

Find the following limit:
lim_(x->infinity) x/(2-cos(x))

Applying the quotient rule, write lim_(x->infinity) x/(2-cos(x)) as (lim_(x->infinity) x)/(lim_(x->infinity) (2-cos(x))):
(lim_(x->infinity) x)/(lim_(x->infinity) (2-cos(x)))

The limit of a difference is the difference of the limits:
(lim_(x->infinity) x)/lim_(x->infinity) 2-lim_(x->infinity) cos(x)

Using the fact that cosine is a continuous function, write lim_(x->infinity) cos(x) as cos(lim_(x->infinity) x):
(lim_(x->infinity) x)/(lim_(x->infinity) 2-cos(lim_(x->infinity) x))

lim_(x->infinity) x  =  infinity:
(lim_(x->infinity) x)/(lim_(x->infinity) 2-cos(infinity))

cos(infinity) = -1 to 1:
(lim_(x->infinity) x)/lim_(x->infinity) 2--1 to 1

Since 2 is constant, lim_(x->infinity) 2  =  2:
(lim_(x->infinity) x)/(2--1 to 1)

2--1 to 1 = 1 to 3:
(lim_(x->infinity) x)/1 to 3

lim_(x->infinity) x  =  infinity:
infinity/(1 to 3)

infinity/(1 to 3) = infinity:
Answer: | infinity

Guest Dec 26, 2015
 #2
avatar+78719 
+8

Look att he graph of the function, here.......https://www.desmos.com/calculator/m8gxiqxa9c

 

Would it be more accurate to say that the limit does not exist.......since the function "ping-pongs"  between values defined by two non-parallel lines as x approaches infinity    ???

 

Anyone else have any thoughts about this question ????

 

 

cool cool cool

CPhill  Dec 26, 2015
 #3
avatar
+10

CPhill: Just plugged it into WolframAlpha and it also gives infinity????!!!!!!!!.

Guest Dec 26, 2015
 #4
avatar+78719 
+5

OK......thanks, guest.....I stand corrected......!!!!

 

 

cool cool cool

CPhill  Dec 26, 2015
edited by CPhill  Dec 27, 2015
 #5
avatar+26329 
+10
Best Answer

The denominator varies between 1 and 3  (finite positive numbers).  The numerator tends to infinity.  Infinity divided by any finite number is still infinite.

Alan  Dec 27, 2015
 #6
avatar+78719 
0

Thanks, Alan......!!!!!

 

 

cool cool cool

CPhill  Dec 27, 2015

6 Online Users

avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details