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Asked Oct 12, 2019
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Good afternoon. I have a midterm coming up, and my professor has given us a few definitions I have never seen before. I cannot seem to make sense of what she is doing, and barely can understand what she is asking for, even when she steps us through them. I was wondering if I could get a plain-english description of what she is looking for, and how to do, the attached question. Thanks!

 

c is continuous at x =
11. Using E-8 definition, prove that the function f(x)
the following table:
e 0.10.01
5. Fill out
0.001
Reminder: Function f(x) is continuous at xo if limzo f(x) = f(x0).
limro f(x)a iffor any e > 0 there exists a 6 > 0 such that for any x if x - xol<8
then \f(x) a< e
help_outline

Image Transcriptionclose

c is continuous at x = 11. Using E-8 definition, prove that the function f(x) the following table: e 0.10.01 5. Fill out 0.001 Reminder: Function f(x) is continuous at xo if limzo f(x) = f(x0). limro f(x)a iffor any e > 0 there exists a 6 > 0 such that for any x if x - xol<8 then \f(x) a< e

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Expert Answer

Step 1

According to the given information it is needed to prove that f(x) is continuous at x=a where

f(x) = x2 and a=5

To prove the following use the definition:

lim f (x)f(x) if for any a 0 there exist 6 0 such that for any x
if -x
x)-f(%)| < e
help_outline

Image Transcriptionclose

lim f (x)f(x) if for any a 0 there exist 6 0 such that for any x if -x x)-f(%)| < e

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Step 2

Now,

let &
0 be given so,
for 8>0 such that x-5|<8= x -25 <s
so, consider 225 - x +5|x-5<6x +5 < 8(10+8)
since, 5(10 6)= e
105-E 0
for instance
-10t /100-4(1))
2
-10 1004
S =
2
since 5>0 so,
S =-5+25+ E
help_outline

Image Transcriptionclose

let & 0 be given so, for 8>0 such that x-5|<8= x -25 <s so, consider 225 - x +5|x-5<6x +5 < 8(10+8) since, 5(10 6)= e 105-E 0 for instance -10t /100-4(1)) 2 -10 1004 S = 2 since 5>0 so, S =-5+25+ E

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Step 3

So, for...

8 =-5+25+ E
lim x2 25
x5
therefore,f(x) x is continuous at x =5
help_outline

Image Transcriptionclose

8 =-5+25+ E lim x2 25 x5 therefore,f(x) x is continuous at x =5

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