Maths with David

IBDP. AA. HL. Differentiation. Limits

Opening Problem

We will define x_n as 0.333333… , with n 3s after the decimal point (e.g. so x₂ = 0.33). n is restricted to positive integers.

n	x_n	3x_n	1 – 3x_n
1
2
3
4
5
10

Copy and complete the table above. In the last column, how many zeroes are there between the decimal point and the 1?
Consider x₁₀₀, which contains 100 3s. In the number 1 – 3x₁₀₀, how many zeroes are there between the decimal point and the 1?
As n gets progressively larger, how many zeroes are there in 1 – 3x_n between the decimal point and the 1? How close does x_n get to 1/3?

The opening problem shows us an example of a limiting process. As n tends to infinity the sequence approaches the limiting value 1/3. Another way of saying this is that for any number we might choose which is close to 1/3, we can find a value of n for which the sequence is closer and remains closer. We write $\lim_{ x \to \infty} = \frac{1}{3}$ .

We will use the following definition of a limit, which is sufficiently accurate for this course:

Definition of Limit: If f(x) is as close as we like to some real number A for all x sufficiently close to (but not equal to) a, then we say that f(x) has a limit of A as x approaches a, and we write $\lim_{ x \to a} f(x) = A$ .

The function may or may not be defined at x=a and this is irrelevant to the definition of the limit of f as x approaches a. We are interested in the behaviour of the function as x gets very close to a.

For instance, consider $f(x) = \frac{5x+x^2}{x}$ as $x \to 0$ . If we substitute 0 in directly we get the meaningless value 0/0. However, if we rewrite the function as $f(x) = \frac{x(5+x)}{x}$ we can see that for values close to zero f(x) = 5+x.

The graph of y=f(x) is the straight line y = x+5 with a hole, or a discontinuity, at (0,5). Even though f(0) doesn’t exist, the limit still exists, so $f(x) \to 5$ as $x \to 0$ . we write $\lim_{ x \to 0} \frac{5x+x^2}{x} = 5$ .

Limit Laws

For functions f(x) and g(x) with $\lim_{ x \to a} f(x) = l$ and $\lim_{ x \to a} g(x) = m$ , where a, l and m are real numbers, the following is true: