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'Limits'에 해당되는 글 3건

2010.12.20 Continuity and One-Sided Limits
2010.12.20 Special Trigonometric Limits
2010.12.20 Finding Limits Graphically, Numerically and Analytically

Continuity and One-Sided Limits

To say that function F is continuous at x=c means that there is no interruption in the graph of at c.

A function f is continuous at c if the following three conditions are met.
1. The function is defined at x=c.
2. The limit of f(x) exitst at x=c.
3. The limit of f(x) exists at x=c and it is equal to f(c).

To understand continuity on a closed interval, it's good to look at a one-sided limit.
The limit from the right means that x approaches c from values greater than c (see Figure 04).
Similarly, the limit from the left means that x approaches c from values less than c (see Figure 05).

Here are examples. f(x)=x^2 and c=4.

The limit from the right at 4 (Figure 04)

The limit from the left at 4 (Figure 05)

A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b),
the function f is continuous from the right at a and continuous from the left at b.

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Special Trigonometric Limits

There is the limit(sinθ/θ, x=0) = 1 that is the special trigonometric limit.

Here is the proof of the limit(sinθ/θ, x=0)=1.

Area of triangle (tanθ/2) ≥ Area of sector (θ/2) ≥ Area of triangle (sinθ/2)
Multiplying each expression by 2/sinθ produces, 1/cosθ ≥ 2/sinθ ≥ 1
and taking reciprocals and reversing the inequalities yields, cosθ ≤ sinθ/θ ≤ 1
Finally, because 1 = limit(x=cosθ, x=0) ≤ limit(x=sinθ/θ, x=0) = limit(x=1, x=0) = 1
(the squeeze theorem), you can get the limit(x=sinθ/θ, x=0)=1

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Finding Limits Graphically, Numerically and Analytically

It is good to use the concept of "approach" to explain limits of functions.

Let's evaluate the limit(x^2, x = 4).

1. Numerical approach
Evaluate the function f(x)=x^2 at several points near x=4 and use the results to estimate
the limit(x^2, x = 4).

x	-3.9	-3.99	-3.999	4	4.001	4.01	4.1
f(x)	15.21	15.9201	15.992001	16	16.008001	16.0801	16.81

------------------------------> <-----------------------------
f(x) approaches 16. f(x) approaches 16.
From the results shown in the table, you can estimate the limit to be 16.

2. Graphical approach
The limit of numerical approach is reinforced by the graph of f (see Figure 01 below).

3. Analytically approach
The limit of analytically approach is using algebra or calculus.
There is the theorem that evaluate the limit(f(x), x=c), and it makes the limit(f(x), x=c)
be same as f(c).
From the theorem, the limit(x^2, x = 4) is 4^2, 16.

There are some limits that fail to exist.
If the function has behavior that differs from the right and from the left on the point,
the limit of the function on that point will not exist (See Figures 02 below).