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It can be shown (but it takes time and effort) that x
n
, n 3
increases and bounded. Hence, it has a finite limit (which is denoted
e in literature). Sometimes this irrational is called the Euler number,
e =2, 71828182 ...
It is an important number (like π, in trigonometry). The function
y = e
x
is called a natural exponential function, log
e
x is called the
natural logarithm of x and is frequently denoted ln x.
2. Limit of a Function
Definition 10
We say that L is the limit of f(x) at x = a if the sequence
f(x
n
) converges to L (assoonasthesequencex
n
has been chosen
to converge to a;herex
n
= a).
We use the notation lim
x→a
f(x)=L or f(x)
x→a
−−→ L.
It was the definition of limit of a function using the sequences. Here
is an alternative:
Definition 11
We write lim
x→a
f(x)=L and say “the limit of f(x) as x tends to
a,equalsL” if we can make the values of f arbitrarily close to L (as
close to L as we like) by taking x to be sufficiently close to a (but
not equal to a).
We can make the last portion even more precise: “if for every
number ε>0 there is a corresponding number δ>0 such that
|f(x) − L| <εwhenever 0 < |x −a| <δ”. The last definition might
be called the “ε, δ” definition of the limit of a function.
To better grasp that fundamental notion, it is strongly recommended
to practice with a graphing calculator. Namely, if ε>0 is given, then
we draw the horizontal lines y = L+ε and y = L−ε and the graph of
f.Iflim
x→a
f(x)=L, then we can find a number δ>0 such that if we
restrict x to lie in the interval (a −δ, a + δ) and take x = a, then the
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