2.2.5sIs there another approach for determining limits at a point?

Gerri, Felix, and Helga live with each other in a small downtown apartment. Individually, their thermostat temperature preferences fluctuate throughout the day. To make matters worse, Helga likes the apartment warmer than Felix and Gerri do, and Gerri prefers cooler temperatures than Felix.

1. On the same set of axes, sketch three graphs that could represent each roommates’ temperature preferences throughout the day. Label each graph with the name of the person it represents.

2. Gerri and Helga rarely agree on a temperature, but it does happen from time to time. When it does, it only lasts for a moment. What can be said about Felix’s temperature preference when Gerri and Helga agree? How do you know?

3. Write an inequality that expresses the relationship between each roommates’ temperature preferences. Use $f(x)$, $g(x)$, and $h(x)$ to represent Felix, Gerri, and Helga respectively.

In the previous problem, Felix’s temperature preference is always sandwiched between Gerri’s and Helga’s. When Gerri’s and Helga’s temperature preferences get closer and closer, Felix’s temperature preference get squeezed!

1. Let $g(x) = -x^2 + 72$ and $h(x) = x^2 + 72$. If $f(x)$ is “squeezed” between $g(x)$ and $h(x)$, what is $\lim\limits _ { x \rightarrow 0 } f ( x)$? How do you know?

2. Assume you want to determine $\lim\limits _ { x \rightarrow a } f ( x)$ algebraically, but do not know how. If you do not know the equation for $f(x)$, describe the conditions needed to evaluate $\lim\limits _ { x \rightarrow a } f ( x)$ using two other functions, $g(x)$ and $h(x)$.

The concept of squeezing a function between two other functions can be useful when determining limits at a point. Consider the function $f ( x ) = x ^ { 2 } \operatorname { sin } ( \frac { 1 } { x } )$.

1. Determine $\lim\limits _ { x \rightarrow 0 } f ( x )$ graphically.

2. To determine $\lim\limits _ { x \rightarrow 0 } f ( x )$ algebraically, you can squeeze $f(x)$ between two other functions, $g(x)$ and $h(x)$, as long as $g(x) ≤ f(x) ≤ h(x)$ on an interval containing $x = 0$, and $\lim\limits _ { x \rightarrow 0 } g ( x ) = \lim\limits _ { x \rightarrow 0 } h ( x)$. The following steps will be helpful in determining $g(x)$ and $h(x)$.

   1. In this situation, the function $f(x)$ is the product of $x^{2}$ and $\sin(\frac{1}{x})$. Start by focusing on $\sin(\frac{1}{x})$. What do you know about $\sin(\frac{1}{x})$ that indicates $–1 ≤\sin(\frac{1}{x})≤ 1$ is a true inequality?

   2. Notice how $\sin(\frac{1}{x})$ is sandwiched between $–1$ and $1$ in the previous problem. How can you use $x^{2}$ to modify the inequality   $–1 ≤\sin(\frac{1}{x})≤ 1$  so that $x^2\sin(\frac{1}{x})$ is represented in the middle? What would the new inequality be?

   3. Use your answer from part (ii) to determine $\lim\limits _ { x \rightarrow 0 } x ^ { 2 } \operatorname { sin } ( \frac { 1 } { x } )$. Which limits did you actually use to determine $\lim\limits _ { x \rightarrow 0 } x ^ { 2 } \operatorname { sin } ( \frac { 1 } { x } )$?

Suppose $f(x) = π \sin(x - 3) + 1$, $g(x) = (x - 4)^3 + 2$, and $h(x) = x^2 - 3x + 1$. Can you use $g(x)$ and $h(x)$ to determine $\lim\limits _ { x \rightarrow 3 } f ( x )$? Explain how you know.