# Enrichment, Extension, and Application

1 point

## Exercises

1. Further exploration of the limit theorems:
1. Let $$f(x) = \dfrac{1}{x - 1}$$ and $$g(x) = \dfrac{4 - 3x}{1 - x}$$
1. Does $$\displaystyle \lim_{x \rightarrow 1} ~ f(x)$$ exist? What about $$\displaystyle \lim_{x \rightarrow 1} ~ g(x)$$?
2. Could you use the limit sum rule to find $$\displaystyle \lim_{x \rightarrow 1} ~ \left( f(x) + g(x) \right)$$?
3. Does $$\displaystyle \lim_{x \rightarrow 1} ~ \left( f(x) + g(x) \right)$$ exist?
2. Let $$f(x) = x^2 + 2x$$ and $$g(x) = \dfrac{1}{x}$$
1. Does $$\displaystyle \lim_{x \rightarrow 0} ~ f(x)$$ exist? What about $$\displaystyle \lim_{x \rightarrow 0} ~ g(x)$$?
2. Could you use the limit product rule to find $$\displaystyle \lim_{x \rightarrow 1} ~ \left( f(x)g(x) \right)$$?
3. Does $$\displaystyle \lim_{x \rightarrow 0} ~ f(x)g(x)$$ exist?
3. The limit sum and limit product theorems have the form
“If A, then B”
where A and B are mathematical statements. The converse of such a theorem is
“If B, then A”
For example, the converse of the limit sum theorem is
“If $$\displaystyle \lim_{x \rightarrow a} ~ \left( f(x) + g(x) \right)$$ exists, then $$\displaystyle \lim_{x \rightarrow a} ~ f(x)$$ exists and $$\displaystyle \lim_{x \rightarrow a} ~ g(x)$$ exists”
What do your results in part a) and part b) above tell you about the converses of the limit sum and limit product theorems?
2. Creating a continuous extension of a function:
1. What is the domain $$\mathcal{D}$$ of the function $$g(x) = \dfrac{x - 3}{\sqrt{5x + 1} - \sqrt{3x + 7}}$$?
2. Determine $$\displaystyle \lim_{x \rightarrow 3} ~ g(x)$$
3. For what value of the constant $$k$$ is the function $f(x) = \begin{cases} g(x), & \text{ if } x \in \mathcal{D} \\ k, & \text{ if } x = 3 \end{cases}$ continuous on its domain?

Comment: The point $$x = 3$$ is called a removable discontinuity of $$g(x)$$ since $$\displaystyle \lim_{x \rightarrow 3} ~ g(x)$$ exists, allowing us to define the continuous function $$f(x)$$.

3. Exploring rational functions: factors, asymptotes, and removable discontinuities:
1. A function $$g(x)$$ is the ratio of two quadratic polynomials. The numerator is $$x^2 - 1$$, and it is known that the graph $$y = g(x)$$ has a vertical asymptote at $$x = 2$$. It is also known that $$g$$ has a removeable discontinuity at $$x = 1$$, and that $$\displaystyle \lim_{x \rightarrow 1} ~ g(x) = -2$$.
1. Construct an equation for the function $$g(x)$$ and evaluate $$\displaystyle \lim_{x \rightarrow \infty} ~ g(x)$$.
2. Sketch the graph of $$y = g(x)$$.
2. Given the function $$f(x) = \dfrac{x^3 + ax^2 + bx + 5}{x^3 + 2x^2 - x - 2}$$, determine whether there are numbers $$a$$ and $$b$$ such that both $$\displaystyle \lim_{x \rightarrow 1} ~ f(x)$$ exists, and $$\displaystyle \lim_{x \rightarrow - 1} ~ f(x)$$ exists. If so, evaluate both limits.
4. Using a transformation to find a limit:
1. Evaluate $$\displaystyle \lim_{x\rightarrow a} \dfrac{x^3 - a^3}{x-a}$$.
2. Now consider $$\displaystyle \lim_{x \rightarrow a} \dfrac{x^{\frac{1}{3}} - a^{\frac{1}{3}}}{x-a}$$. Let $$u = x^{\frac{1}{3}}$$, and $$b = a^{\frac{1}{3}}$$, and hence transform this to a limit in $$u$$ and $$b$$ which you can evaluate. Express your answer in terms of $$a$$.
3. Sketch a graph of the function $$y = f(x) = x^{\frac{1}{3}}$$ and pick a point $$(x, x^{\frac{1}{3}})$$ on the graph for some $$x \gt 1$$, but near to $$1$$. Suppose that $$a = 1$$ in part b). Hence interpret $$\displaystyle \lim_{x \rightarrow 1} ~ \dfrac{x^{\frac{1}{3}} - 1}{x-1}$$ both geometrically, and as a rate of change.
5. Exploring a difference of infinities:

Consider the function $$f(x) = \sqrt{x^2 + 1} - x$$.

1. Make an educated guess about the behaviour of $$f(x)$$ as $$x \rightarrow \infty$$ and as $$x \rightarrow - \infty$$. Justify your reasoning.
2. Evaluate $$\displaystyle \lim_{x \rightarrow \infty} ~ f(x)$$.
6. A trigonometric limit:

A surveyor takes sightings of the top, $$P$$, of a tree, with elevations $$\theta, 2\theta$$, and $$3\theta$$, as shown.

1. Use geometry and the sine law to show that $\frac{AB}{BC} = \frac{\sin{(3\theta)}}{\sin{(\theta)}}$
2. Write $$\sin{(3\theta)} = \sin{(\theta + 2\theta)}$$, and then use a trigonometric identity and limit laws to show that $$\displaystyle \lim_{\theta \rightarrow 0}~\frac{\sin{(3\theta)}}{\sin{(\theta)}} = 3$$
7. A function $$f$$ is defined by $f(x) = \begin{cases} \dfrac{\large \left \lvert x \right \rvert - \left \lvert x - 2 \right \rvert}{\large x - 1} & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \\ \end{cases}$
1. Does $$\displaystyle \lim_{x \rightarrow 1} ~ f(x)$$ exist? Explain your answer.
2. Is $$f$$ continuous at $$x = 1$$? Why, or why not?
3. Sketch the graph of $$f(x)$$.
8. A surprising limit!

In the diagram, $$C_1$$ is a fixed circle $$(x-a)^2 + y^2 = a^2$$, of radius $$a$$, and $$C_2$$ is a smaller circle $$x^2 + y^2 = u^2$$, with variable radius $$u$$ such that $$0 \lt u \lt a$$. The point $$P$$ is $$(0, u)$$, and $$Q$$ is the first quadrant intersection of $$C_1$$ and $$C_2$$. The line through $$P$$ and $$Q$$ meets the $$x$$-axis at $$R~(f(u), 0)$$.

1. Show that, in terms of the variable $$u$$ and the constant $$a$$, $$f(u) = \dfrac{u^2}{2a-\sqrt{4a^2 -u^2}}$$
2. Hence evaluate $$\displaystyle \lim_{u \rightarrow 0} ~ f(u)$$.
9. Functions of the form $$f(x) = \dfrac{ax}{b+cx}$$ have many applications in the sciences:
1. Suppose that the size of a population at time $$t\geq 0$$ is given by $$N(t)=\dfrac{at}{b+t}$$, where $$a$$ and $$b$$ are positive constants.
1. Determine the time $$t_h$$ at which $$\,N(t_h ) = \frac{1}{2} a\,$$.
2. Given that $$\displaystyle \lim_{t \rightarrow \infty} ~ N(t) = 2.4 \times 10^4$$, find the value of $$a$$. Hence interpret the role of the time $$t_h$$ found in part i).
2. In a predator-prey interaction, the function $$g(x) = \dfrac{ax}{1+cx}$$ represents the number of encounters of prey ($$x$$) per predator during the hunting period of the predators, where $$a$$ and $$c$$ are positive constants.
1. What happens to the encounters per predator as the number of prey, $$x$$, becomes very large (i.e. what is $$\displaystyle \lim_{x \rightarrow \infty} ~ g(x)$$)?
2. The constant $$c$$ in this model is proportional to the time the predator spends eating each prey. What effect does an increase in $$c$$ have on your answer in part i)? Explain why this is reasonable biologically.
3. The velocity $$v$$ of a chemical reaction involving enzymes is described by the Michaelis-Menten equation, $$v = \dfrac{ax}{b+x}$$, where $$x$$ is the substrate concentration. In light of your results from part a), explain the roles of the positive constants $$a$$ and $$b$$ in this model.
10. The Heaviside function $$H$$ is defined by $H(t) = \begin{cases} 0 & \text{if } t \lt 0, \\ 1 & \text{if } t \geq 0. \end{cases}$ It is helpful in representing a physical phenomenon which is ‘switched on’ at $$t = 0$$.
1. Sketch the function $$H( t - \pi )$$ and show that $$\displaystyle \lim_{t \rightarrow \pi} ~ H(t - \pi)$$ does not exist.
2. Write out a definition of the function $$g(t) = H(t) + H(t - \pi )$$. (It will have three ‘pieces’.)
3. Define the function $$f(t) = g(t)\cos{(t)}$$, and sketch the graph of $$y = f(t)$$ on the interval $$[-\pi,3\pi]$$, indicating any points where $$f$$ is discontinuous. Explain your reasoning using the definition of continuity.
11. A “functional” equation to solve:

A function $$f(x)$$ satisfies the equation $$f(x) + xf(1 - x) = 1 + x^2$$.

1. Determine an explicit equation for $$f(x)$$ in terms of $$x$$.
2. Show that $\lim_{x \rightarrow \infty} ~ (f(x) - (2-x)) = 0$ that is, that $$f(x)$$ has an oblique asymptote $$y = 2 - x$$ as $$x \rightarrow \infty$$.
12. An interesting additional tool for evaluating limits:

Consider the limit $$\displaystyle \lim_{x \to 0} ~ \left\lvert x \right\rvert \sin{\left( \frac{\pi}{x} \right)}$$. We cannot use the limit product rule to evaluate the limit since $$\displaystyle \lim_{x \to 0} ~ \sin{\left( \frac{\pi}{x} \right)}$$ does not exist.

Looking at the plot of the function on the interval $$\left[ -\frac{3}{2}, \frac{3}{2} \right]$$ notice that the oscillations grow increasingly more crowded as $$x$$ approaches $$0$$.

Zooming in on $$x = 0$$ by decreasing the width of the interval to $$[-1, 1]$$ gives:

Decreasing the width of the interval to $$\left[ -\frac{1}{2}, \frac{1}{2} \right]$$:

Decreasing the width of the interval to $$\left[ -\frac{1}{10}, \frac{1}{10} \right]$$:

Observe that, no matter how small the viewing interval is, the function continues to oscillate extremely frequently as $$x \to \infty$$. However, the amplitude of the oscillations appear to decrease linearly as $$x \to 0$$, as if the function values are bound, or “hemmed in” by lines of unit slope (observe that the amplitude approximately equals the magnitude of the $$x$$ value, implying a slope of $$1$$ on the bounding lines). This appears to force the limit to $$0$$ as $$x \to 0$$.

This behaviour is made more apparent by combining the function with the apparent bounds $$\left\lvert x \right\rvert$$ and $$-\left\lvert x \right\rvert$$ on the intervals $$\left[ -1, 1 \right]$$ (left) and $$\left[ -\frac{1}{10}, \frac{1}{10} \right]$$ (right).

It is clear that the functions $$\left\lvert x \right\rvert$$ and $$-\left\lvert x \right\rvert$$ are acting to “funnel” the function $$\left\lvert x \right\rvert \sin{\left( \frac{\pi}{x} \right)}$$ towards a limit of $$0$$ as $$x \to 0$$. This is a result of the fact that the values of the sine function lie between $$1$$ and $$-1$$ for any value of the argument; that is,

$$-1 \leq \sin{\left( \tfrac{\pi}{x} \right)} \leq 1$$ for all $$x \neq 0$$

Thus, multiplying the inequality by $$\left\lvert x \right\rvert$$ (which is non-negative), we see that

$$-\left\lvert x \right\rvert \leq \left\lvert x \right\rvert \sin{\left( \tfrac{\pi}{x} \right)} \leq \left\lvert x \right\rvert$$ for all $$x \neq 0$$

Since both of the “outer” functions $$\pm \left\lvert x \right\rvert$$ approach $$0$$ as $$x \to 0$$, so also does the “inner” function $$\left\lvert x \right\rvert \sin{\left( \frac{\pi}{x} \right)}$$. Informally, the inner function gets “squeezed” or “sandwiched” between the two outer functions.

The above example illustrates the squeeze theorem, which essentially says that

If $$m(x)$$, $$f(x)$$, and $$M(x)$$ are all defined near $$x = a$$, and $m(x) \leq f(x) \leq M(x)$ near $$x = a$$, with $\lim_{x \to a} ~ m(x) = L = \lim_{x \to a} ~ M(x)$ then, we also have $$\displaystyle \lim_{x \to a} ~ f(x) = L$$.
1. Use the squeeze theorem to show that $$\displaystyle \lim_{x \rightarrow 0} ~ x^2 \cos \left(\frac{2}{x}\right) = 0$$.
2. The squeeze theorem can also be applied as $$x \rightarrow \infty$$. Given that $\frac{3x-1}{x} \lt f(x) \lt 3 - \frac{1}{1+x^2} \text{for all } x \gt 10$ what is $$\displaystyle \lim_{x \rightarrow \infty} ~ f(x)$$?