Expert: Paul Klarreich - 5/4/2009

Question
QUESTION: piecewise function is:

f(x)= (4/(e^x-1)   if x<=1
      
      sqrt(3x^2+1)/(x-2)  if x>1



a) lim f(x)=      
  x->0-

b) lim f(x)=
  x-> 1+

c) lim f(x)=
  x->1

d) lim f(x)=
  x-> -INFINITY

e) lim f(x)=
  x-> INFINITY


Not sure how to do because of the e^x. Could you run me through on how to solve these with e^x and apply answers as well so I can figure out similar problems knowing I have the correct answers.

ANSWER: Questioner:   ray
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  limits
Question:  piecewise function is:

f(x)= (4/(e^x-1)   if x<=1
     
     sqrt(3x^2+1)/(x-2)  if x>1

a) lim f(x)=      
 x->0-

b) lim f(x)=
 x-> 1+

c) lim f(x)=
 x->1

d) lim f(x)=
 x-> -INFINITY

e) lim f(x)=
 x-> INFINITY


Not sure how to do because of the e^x. Could you run me through on how to solve these with e^x and apply answers as well so I can figure out similar problems knowing I have the correct answers.
..............................
When you have a 'piecewise definition', and you are interested in

lim  f(x)
x->a

If your 'a' is on one side of the function, just forget about the other side.

But if your 'a' is the 'split point', then:
................
If you want

lim  f(x)
x->a+

then use the right side, the  x > a part.
...................
If you want

lim  f(x)
x->a-

then use the left side, the  x < a part.
..................
If you want

lim  f(x)
x->a

then you must evaluate it using both 'sides'.  If they come out the same, great -- that is the limit.

If not, too bad -- the limit does not exist.
..................

See what you can do with this.  If you get stuck, let me know and I'll see if I can help.


---------- FOLLOW-UP ----------

QUESTION: well for part c of my question i tried doing

4/(e^1-1) = 4/INF(+) = +/+ = INFINITY
its not right though. how do I get the answer.

for part d,
i did 4/-INF = 0

however thats wrong too.

e) I did the same thing by plugging in INF but somehow none of these are correct.


-its already past due, so could you provide answers so I  know how to get to the correct answer and also provide the steps.. mainly for part c,d,& e.
-Also when i have e^x in a problem how do I find the limits, because the only thing i can think of is plugging in whatever value for x. I don't think thats right though.

Answer
Questioner:   ray
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  limits
Question:  QUESTION: piecewise function is:

f(x)= (4/(e^x-1)   if x<=1
     
     sqrt(3x^2+1)/(x-2)  if x>1



a) lim f(x)=      
 x->0-

b) lim f(x)=
 x-> 1+

c) lim f(x)=
 x->1

d) lim f(x)=
 x-> -INFINITY

e) lim f(x)=
 x-> INFINITY


Not sure how to do because of the e^x. Could you run me through on how to solve these with e^x and apply answers as well so I can figure out similar problems knowing I have the correct answers.

ANSWER: Questioner:   ray
Country:  United States
Category:  Calculus
Private:  No

Subject:  limits
Question:  piecewise function is:

f(x)= (4/(e^x-1)   if x<=1
    
    sqrt(3x^2+1)/(x-2)  if x>1

a) lim f(x)=      
x->0-

b) lim f(x)=
x-> 1+

c) lim f(x)=
x->1

d) lim f(x)=
x-> -INFINITY

e) lim f(x)=
x-> INFINITY


Not sure how to do because of the e^x. Could you run me through on how to solve these with e^x and apply answers as well so I can figure out similar problems knowing I have the correct answers.
..............................
When you have a 'piecewise definition', and you are interested in

lim  f(x)
x->a

If your 'a' is on one side of the function, just forget about the other side.

But if your 'a' is the 'split point', then:
................
If you want

lim  f(x)
x->a+

then use the right side, the  x > a part.
...................
If you want

lim  f(x)
x->a-

then use the left side, the  x < a part.
..................
If you want

lim  f(x)
x->a

then you must evaluate it using both 'sides'.  If they come out the same, great -- that is the limit.

If not, too bad -- the limit does not exist.
..................

See what you can do with this.  If you get stuck, let me know and I'll see if I can help.


---------- FOLLOW-UP ----------

QUESTION: well for part c of my question i tried doing

4/(e^1-1) = 4/INF(+) = +/+ = INFINITY
its not right though. how do I get the answer.

for part d,
i did 4/-INF = 0

however thats wrong too.

e) I did the same thing by plugging in INF but somehow none of these are correct.


-its already past due, so could you provide answers so I  know how to get to the correct answer and also provide the steps.. mainly for part c,d,& e.
-Also when i have e^x in a problem how do I find the limits, because the only thing i can think of is plugging in whatever value for x. I don't think thats right though.
...................................................
Hi, Ray,

e^x is just the exponential function.  I hope your teacher has covered it, or you have read about it.  It is continuous everywhere, so....

............................
lim   f(x) =  [limit from the right, or  x > 1]
x->1+
         4
lim    ------- =    << note, 1+ is no longer needed.
x->1   e^x - 1

  4         4
-------- = -----
e^1 - 1    e - 1

That's done.
...........................

lim   f(x)=    [limit from the left, or  x < 1]
x->1-
  
    sqrt(3x^2+1)
lim  ------------- =
x->1   (x-2)


sqrt(3(1)^2+1)
------------- =
  (1-2)


sqrt(3+1)    
--------- = - 2
  -1

.......................
Now those two limits are not the same; so

lim   f(x) does not exist.
x->1


----------------------------------
d) lim f(x)   =    [limit on the left, or  x < 1]
x-> -INFINITY
         sqrt(3x^2+1)
lim         ---------------
x-> -inf        x - 2

Now you have to do some algebra, which is standard for this type of fractional expression.  You 'complexify' the fraction:  Divide top and bottom by x:  (inside sqrt() it is x^2:


         sqrt(3+1/x^2)
lim         ---------------
x-> -inf        1 - 2/x


sqrt(3)
--------- = sqrt(3)
  1

Note: It seemed to take you a long time to read my answer -- four days and only after a reminder.  

So in the future, if you send me questions, (no problem with that) I will wait until I see you have read the last answer before sending the next one.