Question

limit

Answer 1

prove that \[\lim_{x \rightarrow a} f(x)\] =L

Answer 2

if and only if \[\lim_{x \rightarrow a} (f(x)-L)\]

Answer 3

does:
\[\lim_{x \rightarrow a}(f(x)-L) = 0\]
?

Answer 4

if so:
\[\lim_{x \rightarrow a}(f(x)-L)=\lim_{x \rightarrow a}f(x) + \lim_{x \rightarrow a}(L)\]

Answer 5

by linearity

Answer 6

the second term is just L
\[\lim_{x \rightarrow a}f(x) - L = 0\]

Answer 7

Sorry, three posts up I made a typo, should be limf(x) - lim(L)

Answer 8

anyway, then through algebra, you get:
\[\lim_{x \rightarrow a}f(x) = L\]

Answer 9

okk

Answer 10

here are some limit properties:
http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx

Answer 11

what about \[\lim_{x \rightarrow a} |f(x)| =|L|\]

Answer 12

then you have:
\[\pm \lim_{x \rightarrow a}f(x) = \pm L\]

Answer 13

Out of that, you get two cases:
\[\lim_{x \rightarrow a}f(x) =L\]
and
\[\lim_{x \rightarrow a}f(x) =-L\]

Answer 14

how to prove left hand side is equal right hand side

Answer 15

list out the four cases

Answer 16

\[\lim_{x \rightarrow a}f(x) = L\]
\[\lim_{x \rightarrow a}f(x) = -L\]
\[-\lim_{x \rightarrow a}f(x) = L\]
\[-\lim_{x \rightarrow a}f(x) = -L\]

Answer 17

the negatives cancel in the last one

Answer 18

and you can just divide by -1 in the second one

Answer 19

my question is prove that \[\lim_{x \rightarrow a} |f(x)| = |L|\] if and only if \[\lim_{x \rightarrow a}( f(x)-L)\]