Continuity, Laws of limits and Squeeze theorem

Continuity at a point

A function $f(x)$ is continuous at a point $a$ if:

$$\lim_{x \to a} f(x) = f(a)$$

which means that there are three conditions that must be met:

1. $f(a)$ is defined
2. $\lim_{x \to a} f(x)$ exists ($\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = L$)
3. $\lim_{x \to a} f(x) = f(a)$

Examples:

1. $f(x) = \frac{|x|}{x}$ is not continuous at $x = 0$ because $\lim_{x \to 0^-} = -1$ and $\lim_{x \to 0^+} = 1$, so $\lim_{x \to 0} \frac{|x|}{x}$ does not exist.

2. $g(x) = \frac{x^2}{x}$ is not continuous at $x = 0$ because $g(0)$ is not defined.

3. $h(x) = x^2$ is continuous at $x = 0$ because $\lim_{x \to 0} x^2 = 0 = h(0)$.

4. $r(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ 2x & \text{if } x > 2 \end{cases}$
   is continuous at $x = 2$ because $\lim_{x \to 2^-} r(x) = 2^2 = 4$ and $\lim_{x \to 2^+} r(x) = 2 \cdot 2 = 4$, so $\lim_{x \to 2} r(x) = 4 = r(2)$.

Types of Discontinuities:

Removable: the limit exists but $f(a)$ is undefined or mis‑defined. Example: $g(x) = \frac{x^2-1}{x-1}$ at $x = 1$.

Jump: left and right limits exist but differ. Example: $h(x) = \begin{cases} 1 & \text{if } x < 0 \\ 0 & \text{if } x \geq 0 \end{cases}$ at $x = 0$.

Infinite: limit grows without bound. Example: $k(x) = \frac{1}{x^2}$ at $x = 0$.

Note: Continuity on an interval means continuity at every point in that interval.

Properties of limits

1. Constant limit

$$\lim_{x \to a} c = c, \text{ where } c \text{ is a constant}$$

2. Identity limit

$$\lim_{x \to a} x = a$$

3. Sum/difference limit

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:

$$\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M$$

$$\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = L - M$$

Caution: $L$ and $M$ must exist and are finite, so this is not true for $\infty - \infty$ or $\infty + (-\infty)$.

4. Product limit

$$\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M$$

Example: $\lim_{x \to 3} (x^2) = \lim_{x \to 3} x \cdot \lim_{x \to 3} x = 3 \cdot 3 = 9$

Caution: $L$ and $M$ must exist and are finite, so this is not true for $\infty \cdot 0$ or $\infty \cdot (-\infty)$.

5. Quotient limit

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M}, \text{ where } M \neq 0$$

Caution: $L$ and $M$ must exist and are finite, so this is not true for $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

6. Power limit

If $\lim_{x \to a} f(x) = L$ and $n$ is any real number, then:

$$\lim_{x \to a} (f(x))^n = (\lim_{x \to a} f(x))^n = L^n$$

You can see that if $n$ is a fraction of the form $\frac{1}{n}$, then the power limit is the $n$-th root of the limit.
So, for example, if $n = \frac{1}{2}$, $\lim_{x \to a} (f(x))^{\frac{1}{2}} = \sqrt{\lim_{x \to a} f(x)} = \sqrt{L}$.

Corollary

If $p(x)$ is a polynomial function, then:

$$\lim_{x \to a} p(x) = p(a)$$

Example: $\lim_{x \to 3} (x^2 + 2x + 1) = 3^2 + 2 \cdot 3 + 1 = 16$

Squeeze theorem

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an open interval containing $a$, except possibly at $a$, and if

$$\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$$

then

$$\lim_{x \to a} g(x) = L$$

Example:

$$\lim_{x \to 0} x \sin(\frac{1}{x}) = 0$$

Reasoning:

For all $x$, $-1 \leq \sin(\frac{1}{x}) \leq 1$.

Multiply through by $x$:

$$-x \leq x \sin(\frac{1}{x}) \leq x$$

Since $\lim_{x \to 0} (-x) = 0$ and $\lim_{x \to 0} x = 0$, by the Squeeze Theorem,

$$\lim_{x \to 0} x \sin(\frac{1}{x}) = 0$$

Exercises

1. Check your understanding - Basic limit properties

2. Applying limit properties step by step

3. Continuity practice

a) Is the following function continuous at $x = 2$?

$$f(x) = \begin{cases} x^2, & x < 2 \\ x+2, & x \geq 2 \end{cases}$$

b) Is the following function continuous at $x = 3$?

$$g(x) = \begin{cases} \frac{x^2-9}{x-3}, & x < 3 \\ 2\sqrt{x+6}, & x \geq 3 \end{cases}$$

c) Is the following function continuous at $x = 0$?

$$h(x) = \begin{cases} 2^{x+1}, & x < 0 \\ x^2 + 1, & x \geq 0 \end{cases}$$

d) Is the following function continuous at $x = 1$?

$$k(x) = \begin{cases} 4^{x-1} - 1, & x < 1 \\ \log_2(x^2), & x \geq 1 \end{cases}$$

e) Is the following function continuous at $x = 1$?

$$l(x) = \begin{cases} \frac{x^2-1}{x+1}, & x < 1 \\ x^3+2x^2-1, & x \geq 1 \end{cases}$$

4. Find continuity values

a) Find $m$ so that the piecewise function below is continuous at $x = 3$:

$$f(x) =\begin{cases} 2x + m, & x < 3 \\ x^2 - 1, & x \geq 3 \end{cases}$$

b) Determine the values of $m, n$ such that the following function is continuous for all real numbers:

$$f(x) =\begin{cases} -x^2 + nx + m, & x < 2 \\ 2x - m, & 2\leq x \leq 4 \\ -x - n, & x > 4 \end{cases}$$

c) Find all $n$ so that the piecewise function below is continuous for all real numbers:

$$f(x) = \begin{cases} 2^x, & x < n \\ x^2, & x \geq n \end{cases}$$

5. Squeeze theorem applications

Use the squeeze theorem to find the following limits:

a) $\lim_{x \to 0} x \cdot 2^{\frac{-1}{x^2}}$

(Hint: for $x \neq 0$, note that $0 < 2^{\frac{-1}{x^2}} < 1$)

b) $\lim_{x \to 0} x^2 \cos(\frac{1}{x})$

6. Graphical exploration of squeeze theorem

Using graphical software (like Desmos or GeoGebra), plot the following functions and identify the limits:

a) Plot the functions $f(x) = -x^2$, $g(x) = x^2 \sin(\frac{1}{x})$, and $h(x) = x^2$ on the same graph. Observe how $g(x)$ is "squeezed" between $f(x)$ and $h(x)$.

b) $f(x) = x \sin(\frac{1}{x})$ as $x$ approaches 0. Verify the squeeze theorem.

c) $g(x) = \frac{\sin(x)}{x}$ as $x$ approaches 0.