Limits and Continuity

Exercise 1

Find the following limits:

1. $\lim_{x \to 1} \frac{x^2 - 1}{x^3 - 1}$
2. $\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$
3. $\lim_{x \to 3} \frac{x^3 - 8}{x^2 - 4}$ (Note the changed limit point)
4. $\lim_{x \to y} \frac{x^n - y^n}{x - y}$
5. $\lim_{y \to x} \frac{x^n - y^n}{x - y}$
6. $\lim_{h \to 0} \frac{\sqrt{a + h} - \sqrt{a}}{h}$

Exercise 2

Find the following limits:

1. $\lim_{x \to 1} \frac{1 - \sqrt{x}}{1 - x}$
2. $\lim_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x}$
3. $\lim_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x^2}$

Exercise 3

In each of the following cases, determine the limit $l$ for the given $a$, and prove that it is the limit by showing how to find a $\delta$ such that $| f(x) - l | < \varepsilon$ where $\forall x$ satisfying $0 < |x-a| < \delta$

1. $f(x) = x[3 - \cos(x^2)], \ a = 0$
2. $f(x) = x^2 + 5x - 2, \ a = 2$
3. $f(x) = \frac{100}{x}, \ a = 1$
4. $f(x) = x^4, \ \text{arbitrary } a$
5. $f(x) = x^4 + \frac{1}{x}, \ a = 1$
6. $f(x) = \frac{x}{2 - \sin^2 x}, \ a = 0$
7. $f(x) = \sqrt{|x|}, \ a = 0$
8. $f(x) = \sqrt{x}, \ a = 1$

Exercise 4

Given functions $f$ and $g$ with properties:

\[ \text{If } 0 < |x - 2| < \sin^2 \left( \frac{\epsilon^2}{9} \right) + \epsilon, \text{ then } |f(x) - 2| < \epsilon \]

\[ \text{If } 0 < |x - 2| < \epsilon^2, \text{ then } |g(x) - 4| < \epsilon \]

For each $\epsilon > 0$, find $\delta > 0$ such that:

1. If $0 < |x - 2| < \delta$, then $|f(x) + g(x) - 6| < \epsilon$
2. If $0 < |x - 2| < \delta$, then $|f(x)g(x) - 8| < \epsilon$
3. If $0 < |x - 2| < \delta$, then $\left| \frac{1}{g(x)} - \frac{1}{4} \right| < \epsilon$
4. If $0 < |x - 2| < \delta$, then $\left| \frac{f(x)}{g(x)} - \frac{1}{2} \right| < \epsilon$

Exercise 5

1. If $\lim\limits_{x\to a}f(x)$ and $\lim\limits_{x\to a}g(x)$ do not exist, can $\lim\limits_{x\to a}[f(x)+g(x)]$ exist? Can $\lim\limits_{x\to a}f(x)g(x)$ exist?
2. If $\lim\limits_{x\to a}f(x)$ exists and $\lim\limits_{x\to a}[f(x)+g(x)]$ exists, must $\lim\limits_{x\to a}g(x)$ exist?
3. If $\lim\limits_{x\to a}f(x)$ exists and $\lim\limits_{x\to a}g(x)$ does not exist, can $\lim\limits_{x\to a}[f(x)+g(x)]$ exist?
4. If $\lim\limits_{x\to a}f(x)$ exists and $\lim\limits_{x\to a}f(x)g(x)$ exists, does it follow that $\lim\limits_{x\to a}g(x)$ exists?

Exercise 6

Prove that $\lim\limits_{x\to a}f(x) = \lim\limits_{h\to 0}f(a+h)$.

Exercise 7

1. Prove that $\lim\limits_{x\to a}f(x) = l$ if and only if $\lim\limits_{x\to a}[f(x)-l] = 0$.
2. Prove that $\lim\limits_{x\to 0}f(x) = \lim\limits_{x\to a}f(x-a)$.
3. Prove that $\lim\limits_{x\to 0}f(x) = \lim\limits_{x\to 0}f(x^3)$.
4. Give an example where $\lim\limits_{x\to 0}f(x^2)$ exists, but $\lim\limits_{x\to 0}f(x)$ does not.

Exercise 8

Suppose there is a $\delta > 0$ such that $f(x) = g(x)$ when $0 < |x-a| < \delta$. Prove that $\lim\limits_{x\to a}f(x) = \lim\limits_{x\to a}g(x)$.

Exercise 9

1. Suppose that $f(x) \leq g(x)$ for all $x$. Prove that $\lim\limits_{x\to a}f(x) \leq \lim\limits_{x\to a}g(x)$, provided these limits exist.
2. How can the hypotheses be weakened?
3. If $f(x) < g(x)$ for all $x$, does it necessarily follow that $\lim\limits_{x\to a}f(x) < \lim\limits_{x\to a}g(x)$?

Exercise 10

Suppose that $f(x) \leq g(x) \leq h(x)$ and that $\lim\limits_{x\to a}f(x) = \lim\limits_{x\to a}h(x)$. Prove that $\lim\limits_{x\to a}g(x)$ exists and equals $\lim\limits_{x\to a}f(x) = \lim\limits_{x\to a}h(x)$.

Exercise 11

1. Prove that if $\lim\limits_{x\to 0}\frac{f(x)}{x} = l$ and $b \neq 0$, then $\lim\limits_{x\to 0}\frac{f(bx)}{x} = bl$.
2. What happens if $b = 0$?
3. Find $\lim\limits_{x\to 0}\frac{\sin 2x}{x}$ using $\lim\limits_{x\to 0}\frac{\sin x}{x}$ and another method.

Exercise 12

Evaluate in terms of $\alpha = \lim\limits_{x\to 0}\frac{\sin x}{x}$:

1. $\lim\limits_{x\to 0}\frac{\sin 3x}{x}$
2. $\lim\limits_{x\to 0}\frac{\sin ax}{\sin bx}$
3. $\lim\limits_{x \to 0} \frac{\sin^2 2x}{x}$
4. $\lim\limits_{x \to 0} \frac{\sin^2 2x}{x^2}$
5. $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2}$
6. $\lim\limits_{x \to 0} \frac{\tan^2 x + 2x}{x + x^2}$
7. $\lim\limits_{x \to 0} \frac{x \sin x}{1 - \cos x}$
8. $\lim\limits_{h \to 0} \frac{\sin(x + h) - \sin x}{h}$
9. $\lim\limits_{x \to 1} \frac{\sin(x^2 - 1)}{x - 1}$
10. $\lim\limits_{x \to 0} \frac{x^2 (3 + \sin x)}{(x + \sin x)^2}$
11. $\lim\limits_{x \to 1} (x^2 - 1)^3 \sin \left( \frac{1}{x - 1} \right)^3$

Exercise 13

1. Prove that if $\lim\limits_{x \to a} f(x) = l$, then $\lim\limits_{x \to a} |f|(x) = |l|$
2. Prove that if $\lim\limits_{x \to a} f(x) = l$ and $\lim\limits_{x \to a} g(x) = m$, then:
   a. $\lim\limits_{x \to a} \max(f, g)(x) = \max(l, m)$
   b. Similarly for min

Exercise 14

1. Prove that $\lim\limits_{x \to 0} 1/x$ does not exist (show $\lim\limits_{x \to 0} 1/x = l$ is false for every $l$)
2. Prove that $\lim\limits_{x \to 1} 1/(x - 1)$ does not exist

Exercise 15

Prove that if $\lim\limits_{x \to a} f(x) = l$, then $\exists \delta > 0$ and $M$ such that $|f(x)| < M$ when $0 < |x - a| < \delta$

Exercise 16

Prove that if:

1. $f(x) = 0$ for irrational $x$
2. $f(x) = 1$ for rational $x$
   Then $\lim\limits_{x \to a} f(x)$ does not exist for any $a$

Exercise 17

Prove that if:

1. $f(x) = x$ for rational $x$
2. $f(x) = -x$ for irrational $x$
   Then $\lim\limits_{x \to a} f(x)$ does not exist if $a \neq 0$

Exercise 18

1. Prove that if $\lim\limits_{x \to 0} g(x) = 0$, then $\lim\limits_{x \to 0} g(x) \sin(1/x) = 0$
2. Generalization: If $\lim\limits_{x \to 0} g(x) = 0$ and $|h(x)| \leq M$ for all $x$, then $\lim\limits_{x \to 0} g(x)h(x) = 0$

Exercise 19

Consider a function $f$ with the property: if $g$ is any function where $\lim\limits_{x \to 0} g(x)$ DNE, then $\lim\limits_{x \to 0} [f(x) + g(x)]$ also DNE. Prove this occurs if and only if $\lim\limits_{x \to 0} f(x)$ exists.

Exercise 20

1. If $\lim\limits_{x \to 0} f(x)$ exists and is $\neq 0$, and $\lim\limits_{x \to 0} g(x)$ DNE, prove $\lim\limits_{x \to 0} f(x)g(x)$ DNE
2. Prove the same when $\lim\limits_{x \to 0} |f(x)| = \infty$
3. If neither condition holds, show there exists $g$ where $\lim\limits_{x \to 0} g(x)$ DNE but $\lim\limits_{x \to 0} f(x)g(x)$ exists

Exercise 21

1. For finite sets $A_n \subset [0,1]$ with $A_n \cap A_m = \emptyset$ ($m \neq n$), define:
   \[ f(x) = \begin{cases} 1/n, & x \in A_n \\ 0, & x \notin \bigcup A_n \end{cases} \]
   Prove $\lim\limits_{x \to a} f(x) = 0$ for all $a \in [0,1]$

Exercise 22

Explain why these are all correct definitions of $\lim\limits_{x \to a} f(x) = l$:

For $\delta > 0$, $\exists \varepsilon > 0$ such that:

1. $0 < |x - a| < \varepsilon \implies |f(x) - l| < \delta$
2. $0 < |x - a| < \varepsilon \implies |f(x) - l| \leq \delta$
3. $0 < |x - a| < \varepsilon \implies |f(x) - l| < 5\delta$
4. $0 < |x - a| < \varepsilon/10 \implies |f(x) - l| < \delta$

Exercise 23

Give counterexamples showing these definitions are incorrect:

1. For all $\delta > 0$, $\exists \varepsilon > 0$ such that $0 < |x - a| < \delta \implies |f(x) - l| < \varepsilon$
2. For all $\varepsilon > 0$, $\exists \delta > 0$ such that $|f(x) - l| < \varepsilon \implies 0 < |x - a| < \delta$

Exercise 24

Prove that $\lim\limits_{x \to a} f(x)$ exists if $\lim\limits_{x \to a^+} f(x) = \lim\limits_{x \to a^-} f(x)$

Exercise 25

Prove the following limit relations:

1. $\lim\limits_{x \to 0^+} f(x) = \lim\limits_{x \to 0^-} f(-x)$
2. $\lim\limits_{x \to 0} f(|x|) = \lim\limits_{x \to 0^+} f(x)$
3. $\lim\limits_{x \to 0} f(x^2) = \lim\limits_{x \to 0^+} f(x)$

Exercise 26

1. If $\lim\limits_{x \to a^-} f(x) < \lim\limits_{x \to a^+} f(x)$, prove $\exists \delta > 0$ such that $f(x) < f(y)$ whenever $x < a < y$ and $|x - a| < \delta$, $|y - a| < \delta$
2. Determine whether the converse holds

Exercise 27

1. Prove $\lim\limits_{x \to \infty} \frac{a_n x^n + \cdots + a_0}{b_m x^m + \cdots + b_0}$ exists iff $m \geq n$ ($a_n, b_m \neq 0$)
2. Determine the limit when:
   a. $m = n$
   b. $m > n$

Exercise 28

Find the following limits:

1. $\lim\limits_{x \to \infty} \frac{x + \sin^3 x}{5x + 6}$
2. $\lim\limits_{x \to \infty} \frac{x \sin x}{x^2 + 5}$
3. $\lim\limits_{x \to \infty} \sqrt{x^2 + x} - x$
4. $\lim\limits_{x \to \infty} \frac{x^2 (1 + \sin^2 x)}{(x + \sin x)^2}$

Exercise 29

Prove $\lim\limits_{x \to 0^+} f(1/x) = \lim\limits_{x \to \infty} f(x)$

Exercise 30

Find in terms of $\alpha = \lim\limits_{x \to 0} \frac{\sin x}{x}$:

1. $\lim\limits_{x \to \infty} \frac{\sin x}{x}$
2. $\lim\limits_{x \to \infty} x \sin \frac{1}{x}$

Exercise 31

1. Define $\lim\limits_{x \to -\infty} f(x) = l$
2. Find $\lim\limits_{x \to -\infty} \frac{a_n x^n + \cdots + a_0}{b_m x^m + \cdots + b_0}$
3. Prove $\lim\limits_{x \to -\infty} f(x) = \lim\limits_{x \to \infty} f(-x)$
4. Prove $\lim\limits_{x \to 0^-} f(1/x) = \lim\limits_{x \to -\infty} f(x)$

Exercise 32

1. Define $\lim\limits_{x \to a} f(x) = \infty$ (with formal $\epsilon-\delta$ definition)
2. Show $\lim\limits_{x \to 3} \frac{1}{(x-3)^2} = \infty$
3. If $f(x) > \epsilon > 0$ for all $x$ and $\lim\limits_{x \to a} g(x) = 0$, prove $\lim\limits_{x \to a} \frac{f(x)}{|g(x)|} = \infty$

Exercise 33

1. Define:
   a. $\lim\limits_{x \to a^+} f(x) = \infty$
   b. $\lim\limits_{x \to a^-} f(x) = \infty$
2. Prove $\lim\limits_{x \to 0^+} \frac{1}{x} = \infty$
3. Prove $\lim\limits_{x \to 0^+} f(x) = \infty \iff \lim\limits_{x \to \infty} f(1/x) = \infty$

Exercise 34

Find these limits when they exist:

1. $\lim\limits_{x \to \infty} \frac{x^3 + 4x - 7}{7x^2 - x + 1}$
2. $\lim\limits_{x \to \infty} x(1 + \sin^2 x)$
3. $\lim\limits_{x \to \infty} x \sin^2 x$
4. $\lim\limits_{x \to \infty} x^2 \sin \frac{1}{x}$
5. $\lim\limits_{x \to \infty} \sqrt{x^2 + 2x} - x$
6. $\lim\limits_{x \to \infty} x(\sqrt{x + 2} - \sqrt{x})$
7. $\lim\limits_{x \to \infty} \frac{\sqrt{|x|}}{x}$

Exercise 35

1. Find perimeter of regular $n$-gon inscribed in circle radius $r$
2. Determine perimeter’s limiting value as $n \to \infty$
3. Identify the fundamental limit this suggests

Exercise 36

1. For $c > 1$, prove $\lim\limits_{n \to \infty} c^{1/n} = 1$
   Hint: Show $c^{1/n} \leq 1 + \epsilon$ for large $n$ and any $\epsilon > 0$
2. Generalize to $c > 0$, prove $\lim\limits_{n \to \infty} c^{1/n} = 1$