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anonymous
 one year ago
use the definition of continuity to determine whether f is continuous at a.
anonymous
 one year ago
use the definition of continuity to determine whether f is continuous at a.

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perl
 one year ago
Best ResponseYou've already chosen the best response.1I can't read that because i don't have microsoft word. one moment

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok thank you so much for helping me

perl
 one year ago
Best ResponseYou've already chosen the best response.1apparently the file to download is huge

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0f(x)= { 2x if x<1 { 1 if x=1 { x^2 if x>1 a=1

perl
 one year ago
Best ResponseYou've already chosen the best response.1for continuity we need to check the points where the function changes, the rest of the function is continuous in its respective parts.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok how do we do that

perl
 one year ago
Best ResponseYou've already chosen the best response.1Definition of continuity: Left hand limit = right hand limit = value at the point

perl
 one year ago
Best ResponseYou've already chosen the best response.1We can write this in symbols.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0you write it with a little apostrophe thing right

perl
 one year ago
Best ResponseYou've already chosen the best response.1Definition of continuity: $$ \Large \lim_{x \to a} f(x) = f(a)$$But this implies $$ \Large \lim_{x \to a^{+}} f(x) = \Large \lim_{x \to a^{}} f(x) = f(a) $$

perl
 one year ago
Best ResponseYou've already chosen the best response.1So we need to check the left limit and right limits and make sure it equals to the value at that point.

perl
 one year ago
Best ResponseYou've already chosen the best response.1$$ \rm \Large{f(x)= \begin{cases} 2x ~~~if~ x<1\\ 1 ~~~~~~~~ ~\rm ~if~ x=1\\ x^2 \rm ~~~~~~~~if~ x>1 \end{cases} } $$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0for all the x's i replace them with a 1?

perl
 one year ago
Best ResponseYou've already chosen the best response.1\[ \rm \Large{f(x)= \begin{cases} 2x ~~~if~ x<1\\ 1 ~~~~~~~~ ~\rm ~if~ x=1\\ x^2 \rm ~~~~~~~~if~ x>1 \end{cases} } \] \[ \Large \lim_{x \to 1^{+}} f(x) = \\~\\ \Large \lim_{x \to 1^{}} f(x) = \\~\\ \Large f(1) = \]

perl
 one year ago
Best ResponseYou've already chosen the best response.1yes thats correct. and we need to check that all three of those are equal . thats the continuity condition

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so the answer is 1 for all of them.

perl
 one year ago
Best ResponseYou've already chosen the best response.1\[ \rm \Large{f(x)= \begin{cases} 2x ~~~if~ x<1\\ 1 ~~~~~~~~ ~\rm ~if~ x=1\\ x^2 \rm ~~~~~~~~if~ x>1 \end{cases} } \] \[ \Large \lim_{x \to 1^{}} f(x) =\lim_{x \to 1^{}} 2x = 21 = 1 \\~\\ \Large \lim_{x \to 1^{+}} f(x) = \lim_{x \to 1^{+}} x^2 = 1^2 = 1 \\~\\ \Large f(1) = 1 \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thank You So Much,can you help me with one more similar to this one.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0determine for what numbers, if any, the given function is discontinuous. f(x)={x+7 if x is less than or equal to 0 {7 if 0 is <x less than or equal to 3 {x^21 if x>3

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok whats the difference between the first set and the second set. (not the questions but what you did)

perl
 one year ago
Best ResponseYou've already chosen the best response.1there are two potential points of discontinuity, where the function abruptly changes. it changes at x =0 and x = 3. so we have to test those two parts independently

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok so how do we do that

perl
 one year ago
Best ResponseYou've already chosen the best response.1We can look at the graph that might make it more clear https://www.desmos.com/calculator/pcjrrf5xdh

perl
 one year ago
Best ResponseYou've already chosen the best response.1edit* \[ \rm \Large{f(x)= \begin{cases} x+7 ~~~if~ x\leq0\\ 7 ~~~~~~~~ ~\rm ~if~ 0 < x\leq 3\\ x^21 \rm ~~if~ x>3 \end{cases} } \] \[ \Large \lim_{x \to 0^{}} f(x) =\lim_{x \to 0^{}} x+7 = 0+7 = 7 \\~\\ \Large \lim_{x \to 0^{+}} f(x) = \lim_{x \to 0^{+}} 7 = 7 \\~\\ \Large f(0) =0+7 = 7 \] \[ \Large \lim_{x \to 3^{}} f(x) =\lim_{x \to 3^{}}7 = 7 \\~\\ \Large \lim_{x \to 3^{+}} f(x) = \lim_{x \to 3^{+}} x^21 = 3^21 = 8 \\~\\ \Large f(3) = 7 \]

perl
 one year ago
Best ResponseYou've already chosen the best response.1from the graph and from the continuity definition we see that there is a point of discontinuity at x = 3. x=0 is fine

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so all the question was asking for was two points

perl
 one year ago
Best ResponseYou've already chosen the best response.1Before we do any work the function is potentially discontinuous at the two points x=0 and x = 3 because the function changes suddenly at those points. But after analyzing the function graphically, or using the limit above, we see that it is actually only discontinuous at one point. x= 3
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