We write:limx → c f(x) = Lorf(x) → Lasx → c
if the functional value f(x) is close to the single real number L whenever x is close, but not equal, to c (on either side of c.)
Note that:
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We write:limx → c- f(x) = K
and call K the limit from the left or the left-hand limit if f(x) is close to K whenever x is close to, but to the left of, c on the real number line.
We write:limx → c+ f(x) = L
and call L the limit from the right or the right-hand limit if f(x) is close to L whenever x is close to, but to the right of, c on the real number line.
For a (two-sided) limit to exist, the limit from the left and the limit from the right must exist and be equal. That is,
limx → c f(x) = Lif and only iflimx → c- f(x) = limx → c+ f(x) = L
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Let f and g be two functions, and assume that:
limx → c f(x) = Landlimx → c g(x) = M
where L and M are real numbers (both limits exist). Then:
limx → ck = k | limx → cx = x |
limx → c[f(x) + g(x)] = limx → cf(x) + limx → cg(x) = L + M | limx → c[f(x) - g(x)] = limx → cf(x) - limx → cg(x) = L - M |
limx → ck f(x) = klimx → cf(x) = kL | limx → c[f(x) ⋅ g(x)] = [limx → cf(x)] [limx → cg(x)] = LM |
limx → c[f(x)/g(x)] = [limx → cf(x)]/[limx → cg(x)] = L/M if M ≠ 0 | limx → cn√ f(x) = n√ limx → c f(x) = n√ L, L > 0 for n even |
then limx → cf(x)/g(x) is said to be indeterminate, or, more specifically, a 0/0 indeterminate form.
If limx → cf(x) = L, L ≠ 0 , and limx → cg(x) = 0
then limx → cf(x)/g(x) does not exist.