12. x lim x
lim = 1
x 0+
x = 1 x 0 x
Right hand limit: lim f(x) = L means that f(x) can be
x a +
made as close to L as we wish by taking x sufficiently close
to a but greater than a
Left hand limit: lim f(x) = L means that f(x) can be
x a
made as close to L as we wish by taking x sufficiently close
to a but less than a
13. lim sinx = lim sinx
+ = lim sinx = 1
x 0
x x 0 x x 0
x
lim x
lim
= x 0
x lim x
therefore, x 0 does not exist
x 0+
x
x x
14. Rules for calculation limits
1. Sum Rule: lim [ f(x) + g(x) ] = lim f(x) + lim g(x) = L + M
x a x a x a
2. Difference Rule: lim [ f(x) g(x) ] = lim f(x) lim g(x)
x a x a x a
= L M
x a
. .
3. Product Rule: lim [ f(x) g(x) ] = lim f(x) lim g(x) = LM
x a x a