IT204 - Limits

Definitions and Properties

Dr Robert Batzinger
Instructor Emeritus

Payap University
Chiang Mai, Thailand
15-Aug-2022

0.1 Agenda

Rates of change
Finding limits graphically and numerically
Evaluating limits analytically
Continuity and one-sided limits
Sandwich theorem
Infinite limits

1 Rates of change

1.1 Average rate of change

Given \(y = f(x)\):

\[\frac{\Delta y}{\Delta x} = \frac{f(x_2)-f(x_1)}{x_2 - x_1} = \frac{f(x_1+h) - f(x_1)}{h},h \ne 0\] Assuming the curve is continuous:

As the step get smaller the value becomes closer to the real value.
Approaching from the left or from the right, closes on the same value.

1.2 Estimating rate of change at a point on the curve

1.3 The effects around a relative maximum or minima

1.4 Behaviour around point of inflection

1.5 Behavoir near discontinuous point

1.6 Behavior near asymptote

1.7 Modeling functions with large \(|x|\)

1.8 Modeling functions with large \(|x|\)

1.9 Angular asymtopes

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1.9.1 Calculations: Polynomial division

\[y = \frac{2x^2-3}{7x+4}\]

\[\begin{matrix} \frac{2x^2-3}{7x+4} &=&\frac{2x}{7} &+& -\frac{8}{49}& &,& R = \frac{-115}{49(7x+4)}\\ Long\ division &\dots &\dots& \dots & \dots &\dots &\dots\\ 7x+4& | & 2x^2 & & &-& 3\\ & | & -2x^2 &+& \frac{8x}{7} & & \\ & |& & & \frac{-8x}{7} &+&\frac{32}{49}\\ & |& & & & &\frac{-115}{49}\\ \end{matrix}\]

1.10 Sandwich between other equations

1.10.1 Another sandwich

\[1-\frac{x^2}{4} \le u(x) \le 1 +\frac{x^2}{2}\] \[\lim_{x\rightarrow 0} \left(1-\frac{x^2}{4}\right) =1, \quad\lim_{x\rightarrow 0} \left(1-\frac{x^2}{2}\right) =1,\] \[\therefore \lim_{x\rightarrow 0} u(x) = 1\]

2 Sandwich theorem

\[Given\ g(x) \le f(x) \le h(x)\]

\[if \ \lim_{x\rightarrow c} g(x) = \lim_{x\rightarrow c} h(x) = L\] \[\therefore f(x) = L\] # Finding limits graphically and numerically

\[\lim_{x\rightarrow a}\frac{x^2 - a^2}{x^4 - a^4} = \lim_{x\rightarrow a}\frac{x^2 - a^2}{(x^2 - a^2)(x^2 + a^2)} = \lim_{x\rightarrow a}\frac{1}{x^2 + a^2}\approx \frac{1}{2a^2}\]

2.1 Definition

\[\lim_{x\rightarrow x_0} f(x) = L\] \[0 \lt |x - x_0| < \delta\quad \Rightarrow\quad |f(x) - L| < \epsilon\]

2.2 Limit Rules

Assuming \(\lim_{x\rightarrow c} f(x) = L,\quad \lim_{x\rightarrow c} g(x) = M\)

Sum Rule: \(\lim_{x\rightarrow c} \left(f(x) + g(x)\right) = L + M\)
Difference Rule: \(\lim_{x\rightarrow c} \left(f(x) - g(x)\right) = L - M\)
Product Rule: \(\lim_{x\rightarrow c} \left(f(x) \times g(x)\right) = L \times M\)
Constant Multiple Rule: \(\lim_{x\rightarrow c} \left(k f(x)\right) = k \times L\)
Quotient Rule: \(\lim_{x\rightarrow c} \left(f(x) / g(x)\right) = L / M; \quad M \ne 0\)
Power Rule: \(\lim_{x\rightarrow c} \left(f(x)\right)^{r/s} = L^{r/s}\)

3 Evaluating limits analytically

\[\lim_{x\rightarrow 5} \left(\frac{x-5}{\color{red}{x^2-25}}\right) = \lim_{x\rightarrow 5} \left(\frac{\color{red}{x-5}}{\color{red}{(x-5)}(x+5)}\right) = \lim_{x\rightarrow 5} \frac{1}{x+5}\]

4 Continuity and one-sided limits

Continuity: left and right sides

\[\lim_{x\rightarrow a} = f(a);\quad \lim_{x\rightarrow b} = f(b)\]

4.1 Asymptotes

5 Discontinuity

6 Infinite limits

\[As\ x \rightarrow \infty\]

Reduce the equation to its simplest terms
The term of the highest power is the most significant and the other terms can be ignored
Change for patterns of oscillations and repeated asymptotes.