Notes for Calculus

Chapter 2 : Limits

Author: Kenneth, S.K. Cheng

Table of Contents

• Notes for Calculus
  • Chapter 2 : Limits
    • Author: Kenneth, S.K. Cheng
  • Table of Contents
    • 2.1 Limit of Functions
    • 2.2 One-Sided Limits
    • 2.3 Calculating Limits
    • 2.4 Analysis of Limits
    • 2.5 Continuity
    • 2.6 Solving Limits Problem with Python
    • 2.7 Exercises

2.1 Limit of Functions

In this chapter, we will be looking at the concept of limits. The concept of limits is important in calculus as it helps us to understand the behavior of functions as they approach a certain value. We will be looking at the definition of limits, the properties of limits, and the different types of limits.

One may wonder why we have to learn about limits. It actually comes from the problem of investigating the behavior of functions as they approach a certain value. For example, consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. If we substitute $x = 1$ into the function, we get $\frac{1^2 - 1}{1 - 1} = \frac{0}{0}$. This is known as the function does not define at $x = 1$. However, one may still want to know the behavior of the function as it approaches $x = 1$. This is where the concept of limits comes in.

For another function, consider the function $f(x) = x^2 - x +2$ which is defined for all $x \in \mathbb{R}$. Suppose we want to investigate the behavior of the function as $x$ approaches $2$, then we may have the following table of value of $f(x)$ as $x$ approaches $2$:

While from the right side of $x = 2$, we have the following table of value of $f(x)$ as $x$ approaches $2$:

From the tables, we can see that as $x$ approaches $2$, the value of $f(x)$ approaches $4$. This is known as the limit of the function $f(x)$ as $x$ approaches $2$ is $4$. We denote this as $\lim_{x \to 2} f(x) = 4$.

For a more general case, we have

Definition (Intuitive Definition): Let $f(x)$ be a function defined on an open interval containing $x = a$ except possibly at $x = a$. We say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, denoted by $\lim_{x \to a} f(x) = L$, if we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ but not equal to $a$.

Roughly speaking, this says that the value of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ but not equal to $a$.

Example: Let $f(x) = x^2 - 1$ be a function defined for all $x \in \mathbb{R}$. Then, $\lim_{x \to 1} f(x) = 0$.

Example: Let $f(x) = \frac{1}{x}$ be a function defined for all $x \in \mathbb{R} \setminus \{0\}$. Then guess the value of $\lim_{x \to 0} f(x)$.

2.2 One-Sided Limits

In the previous section, we have defined the limit of a function as $x$ approaches a certain value. However, we may also want to investigate the behavior of the function as $x$ approaches the value from the left side or the right side. This is where the concept of one-sided limits comes in.

Definition (One-Sided Limits): Let $f(x)$ be a function defined on an open interval containing $x = a$ except possibly at $x = a$. We say that the limit of $f(x)$ as $x$ approaches $a$ from the left is $L$, denoted by $\lim_{x \to a^-} f(x) = L$, if we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ but not equal to $a$ and $x < a$.

Similarly, we say that the limit of $f(x)$ as $x$ approaches $a$ from the right is $L$, denoted by $\lim_{x \to a^+} f(x) = L$, if we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ but not equal to $a$ and $x > a$.

Remark: $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$.

Example: Discuss whether $\lim_{x \to 0} \frac{1}{x^2}$ exists.

Definition (Infinite Limits): Let $f(x)$ be a function defined on an open interval containing $x = a$ except possibly at $x = a$. We say that the limit of $f(x)$ as $x$ approaches $a$ is $\infty$, denoted by $\lim_{x \to a} f(x) = \infty$, if we can make the values of $f(x)$ arbitrarily large by taking $x$ sufficiently close to $a$ but not equal to $a$.

Similarly, we say that the limit of $f(x)$ as $x$ approaches $a$ is $-\infty$, denoted by $\lim_{x \to a} f(x) = -\infty$, if we can make the values of $f(x)$ arbitrarily large negative by taking $x$ sufficiently close to $a$ but not equal to $a$.

2.3 Calculating Limits

In this section, we will be looking at the methods of calculating limits. There are several methods to calculate limits, such as direct substitution, factoring, rationalizing, and using the squeeze theorem.

But first, we shall provide the properties of limits.

Properties of Limits:

1. Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.
2. Difference Rule: $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$.
3. Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$.
4. Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ if $\lim_{x \to a} g(x) \neq 0$.
5. Scalar Rule: $\lim_{x \to a} c \cdot f(x) = c \cdot \lim_{x \to a} f(x)$.
6. Power Rule: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$.

Example: Calculate $\lim_{x \to 2} (x^2 - 1)$.

Example: Calculate $\lim_{x \to 0} (2x^2 - 3x + 1)$.

Direct Substitution: If $f$ is a polynomial or a rational function and $a$ is in the domain of $f$, then $\lim_{x \to a} f(x) = f(a)$.

Example: Calculate $\lim_{x \to 1} \frac{x^2-1}{x-1}$.

Factoring: If $f(x)$ is a rational function and $f(a)$ is undefined, then we can try to factorize $f(x)$ and simplify the expression.

2.4 Analysis of Limits

Even though we have the intuition of limits, we may still want to investigate the behavior of functions as they approach a certain value. In this section, we will be providing a more rigorous definition of limits. (I just want you to learn more only, it should be contained in your undergraduate course)

Definition (Epsilon-Delta Definition): Let $f(x)$ be a function defined on an open interval containing $x = a$ except possibly at $x = a$. We say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, denoted by $\lim_{x \to a} f(x) = L$, if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

Example: Use the Epsilon-Delta Definition to prove that $\lim_{x \to 1} x^2 = 1$.

Proof: Since $|x^2 - 1| = |x - 1||x + 1|$, we want to find a $\delta$ such that if $0 < |x - 1| < \delta$, then $|x + 1| < \epsilon$. Let $\delta = \min\left\{1, \frac{\epsilon}{2}\right\}$. Then, if $0 < |x - 1| < \delta$, we have $|x + 1| < 2$ and $|x - 1| < 1$. Hence, $|x^2 - 1| = |x - 1||x + 1| < 2|x - 1| < 2\delta = \epsilon$. Hence, $\lim_{x \to 1} x^2 = 1$.

2.5 Continuity

In this section, we will be looking at the concept of continuity. The concept of continuity is an extension of the concept of limits. We know from secondary school that a function is continuous if we can draw the graph of the function without lifting the pen. In this section, we will be providing a more rigorous definition of continuity.

Definition (Continuity): Let $f(x)$ be a function defined on an open interval containing $x = a$. We say that $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

Definition (Continuity on an Interval): Let $f(x)$ be a function defined on an interval $I$. We say that $f(x)$ is continuous on $I$ if $f(x)$ is continuous at every point in $I$.

Example: Prove that the function $f(x) = x^2$ is continuous for all $x \in \mathbb{R}$.

Proof: Let $a \in \mathbb{R}$. Then, $\lim_{x \to a} x^2 = a^2 = f(a)$. Hence, $f(x) = x^2$ is continuous for all $x \in \mathbb{R}$.

Example: Prove that the function $f(x) = \frac{1}{x}$ is continuous for all $x \in \mathbb{R} \setminus \{0\}$.

Proof: Let $a \in \mathbb{R} \setminus \{0\}$. Then, $\lim_{x \to a} \frac{1}{x} = \frac{1}{a} = f(a)$. Hence, $f(x) = \frac{1}{x}$ is continuous for all $x \in \mathbb{R} \setminus \{0\}$.

Properties of Continuous Functions: If $f(x)$ and $g(x)$ are continuous at $x = a$, and if $c$ is a constant, then

1. $f(x) + g(x)$ is continuous at $x = a$.
2. $f(x) - g(x)$ is continuous at $x = a$.
3. $f(x) \cdot g(x)$ is continuous at $x = a$.
4. $\frac{f(x)}{g(x)}$ is continuous at $x = a$ if $g(a) \neq 0$.
5. $c \cdot f(x)$ is continuous at $x = a$.
6. $f(x)^n$ is continuous at $x = a$.
7. $f(g(x))$ is continuous at $x = a$ if $g(x)$ is continuous at $x = a$ and $f(x)$ is continuous at $g(a)$.
8. $\sqrt{f(x)}$ is continuous at $x = a$ if $f(x) \geq 0$ for all $x$ in a neighborhood of $a$.
9. $|f(x)|$ is continuous at $x = a$.

Example: Find the limit $\lim_{x \to -2} \frac{x^3+2x^2-1}{5-3x}$ and determine whether the function is continuous at $x = -2$.

There is an important theorem that states that if $f(x)$ is continuous on $[a,b]$, and let $N$ be a number between $f(a)$ and $f(b)$, then there exists a number $c$ in $[a,b]$ such that $f(c) = N$. This is known as the Intermediate Value Theorem.

Intermediate Value Theorem: Let $f(x)$ be a function defined on a closed interval $[a,b]$. If $f(x)$ is continuous on $[a,b]$, and if $N$ is a number between $f(a)$ and $f(b)$, then there exists a number $c$ in $[a,b]$ such that $f(c) = N$.

Example: Prove that the equation $x^3 - 3x + 1 = 0$ has a solution in the interval $[0,1]$.

Proof: Let $f(x) = x^3 - 3x + 1$. Then, $f(0) = 1$ and $f(1) = -1$. Since $f(x)$ is continuous on $[0,1]$, and $0$ is between $f(0)$ and $f(1)$, by the Intermediate Value Theorem, there exists a number $c$ in $[0,1]$ such that $f(c) = 0$. Hence, the equation $x^3 - 3x + 1 = 0$ has a solution in the interval $[0,1]$.

2.6 Solving Limits Problem with Python

In this section, we will be looking at how to solve limits problem with Python. We will be using the sympy library to solve limits problem.

Example: Find the limit $\lim_{x \to 1} \frac{x^2-1}{x-1}$.

import sympy as sp

x = sp.symbols('x')
f = (x**2 - 1)/(x - 1) # We use ** for exponentiation
limit = sp.limit(f, x, 1) # the first argument is the function, the second argument is the variable, and the third argument is the value the variable approaches
print(limit)

Example: Find the limit $\lim_{x \to 0} \frac{\sin(x)}{x}$.

import sympy as sp

x = sp.symbols('x')
f = sp.sin(x)/x
limit = sp.limit(f, x, 0)
print(limit)

The result of the first example is $2$ and the result of the second example is $1$. The first example is quite obvious, but the latter one is not. We can use Python to verify the result.

2.7 Exercises

1. Find the limit of the following and determine whether the function is continuous at the given value:
   1. $\lim_{x \to 0} \cos(x+\sin x)$
   2. $\lim_{x \to -3} \frac{x^2-9}{x^2+2x-3}$
   3. $\lim_{x \to 3} \frac{x^2-9}{x^2+2x-3}$
   4. $\lim_{r \to 9} \frac{\sqrt{r}-3}{r-9}$
   5. $\lim_{t \to +\infty} \frac{\sqrt{x+\sqrt{x+\sqrt{x+1}}}}{\sqrt{x}}$
   6. $\lim_{x \to +\infty} \sqrt{x+\sqrt{x+\sqrt{x}}} - \sqrt{x}$
2. Use the epsilon-delta definition to prove the following:
   1. $\lim_{x \to 2} (14-5x) = 4$
   2. $\lim_{x \to 3} (x^2-9) = 0$
3. Use the Intermediate Value Theorem to prove the following equations with a solution in the given interval:
   1. $x^5 - x^3 + 3x - 5 = 0$ for $x \in [1,2]$
   2. $2\sin x = 3 - 2x$ for $x \in [0,1]$