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12
2.33
How to find a nonvertical asymptote for the
curve y = f (x) as x → ∞:
• Examine lim f (x)/x . If the limit does not
x→∞
exist, there is no asymptote as x → ∞.
• If lim f (x)/x = a, examine the limit
x→∞
lim f (x) − ax . If this limit does not exist,
x→∞
the curve has no asymptote as x → ∞.
• If lim f (x) − ax = b, then y = ax + b is an
Method for finding nonvertical asymptotes for
a curve y = f (x) as
x → ∞. Replacing
x → ∞ by x → −∞
gives a method for finding nonvertical asymptotes as x → −∞.
x→∞
asymptote for the curve y = f (x) as x → ∞.
To find an approximate root of f (x) = 0, define
xn for n = 1, 2, . . . , by
2.34
xn+1 = xn −
f (xn )
f (xn )
If x0 is close to an actual root x∗ , the sequence
{xn } will usually converge rapidly to that root.
Newton’s approximation method. (A rule of
thumb says that, to obtain an approximation
that is correct to n decimal places, use Newton’s
method until it gives the
same n decimal places
twice in a row.)
y
2.35
x∗
xn xn+1
x
y = f (x)
Illustration of Newton’s
approximation method.
The tangent to the
graph of f at (xn , f (xn ))
intersects the x-axis at
x = xn+1 .
2.36
Suppose in (2.34) that f (x∗ ) = 0, f (x∗ ) = 0,
and that f (x∗ ) exists and is continuous in a
neighbourhood of x∗ . Then there exists a δ > 0
such that the sequence {xn } in (2.34) converges
to x∗ when x0 ∈ (x∗ − δ, x∗ + δ).
Sufficient conditions for
convergence of Newton’s
method.
2.37
Suppose in (2.34) that f is twice differentiable
with f (x∗ ) = 0 and f (x∗ ) = 0. Suppose further that there exist a K > 0 and a δ > 0 such
that for all x in (x∗ − δ, x∗ + δ),
|f (x)f (x)|
≤ K|x − x∗ | < 1
f (x)2
Then if x0 ∈ (x∗ − δ, x∗ + δ), the sequence {xn }
in (2.34) converges to x∗ and
n
|xn − x∗ | ≤ (δK)2 /K
A precise estimation of
the accuracy of Newton’s
method.