The Sandwich Theorem or Squeeze Play Theorem and related problem set

Nalin Pithwa

Dec 28

Reference: Calculus and Analytic Geometry 9th edition, George B. Thomas, Jr. and Ross L. Finney; available Amazon India.

Suppose that $g(x) \leq f(x) \leq h(x)$ for all x in some open interval containing c, except possibly at $x=c$ itself. Suppose also that: $\lim_{x \rightarrow c} g(x) = \lim_{x \rightarrow c} h(x) =L$ Then, $\lim_{x \rightarrow c} f(x) = L$ .

Problems to practice based on above theorem:

Question 1: Given that $1- \frac{x^{2}}{4} \leq u(x) \leq 1 + \frac{x^{2}}{2}$ for all $x \neq 0$ . F, find $\lim_{x \rightarrow 0} u(x)$ .

Question 2: Show that if $\lim_{x \rightarrow c} |f(x)| =0$ , then $\lim_{x \rightarrow c} f(x) = 0$ .

Question 3: If $\sqrt{5-2x^{2}} \leq f(x) \leq \sqrt{5-x^{2}}$ for $-1 \leq x \leq 1$ , find $\lim_{x \rightarrow 0} f(x)$

Question 4: If $2 - x^{2} \leq g(x) \leq 2 \cos {x}$ for all x, find $\lim_{x \rightarrow 0} g(x)$ .

Question 5 (a): It can be shown that the inequalities $1 - \frac{x^{2}}{6} < \frac {x \sin {x}}{2-2\cos {x}} < 1$ hold for all values of x close to zero. What, if anything, does this tell you about the following limit:

$\lim_{x \rightarrow 0} \frac{x \sin {x}}{2-2\cos {x}}$ ?

Give reasons for your answer.

Question 5b: Graph $y = 1 - \frac{x^{2}}{6}$ , $y = \frac {(x \sin {x})}{2 - 2 \cos{x}}$ and $y = 1$ together for $-2 \leq x \leq 2$ . Comment on the behaviour of the graphs as $x \rightarrow 0$ . NB: we can use graphing tools like http://www.desmos.com or http://www.geogebra.com or a nice programmable TI Aspire graphing and programming calculator. (BTW, such nice TI calculators are available in Amazon India also).

Question 6: Suppose that the inequalities:

$\frac{1}{2} - \frac{x^{2}}{24} < \frac{1 - \cos {x}}{x^{2}} < \frac{1}{2}$

hold for values of x close to zero. What, if anything does this tell you about $\lim_{x \rightarrow 0} \frac{ 1- \cos {x}}{x^{2}}$ ?

Give reasons for your answer.

Question 7: Graph the equations $y = (1/2) - (\frac{x^{2}}{24})$ , and $y=\frac{(1-\cos{x})}{x^{2}}$ and $y=\frac{1}{2}$ together for the domain $-2 \leq x \leq 2$ . Comment on the behaviour of the graphs as $x \rightarrow 0$ .