Algebraic manipulation of limits and asymptotes

Some Algebraic Rules

$\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$
$a^2 - b^2 = (a+b)(a-b)$
$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
$a^4 - b^4 = (a+b)(a-b)(a^2 + b^2)$
$ax^2 + bx + c = a(x-r_1)(x-r_2)$ where $r_1$ and $r_2$ are the roots of the quadratic equation $ax^2 + bx + c = 0$

Usually, you will see these rules involving a number in place of $a$ or $b$ .
For example:

$x^2 - 4 = (x-2)(x+2)$

$x^2 - 1 = (x-1)(x+1)$

$x^3 - 1 = (x-1)(x^2 + x + 1)$

$x^3 + 1 = (x+1)(x^2 - x + 1)$

For quadratic expressions, you can use the quadratic formula to find the roots.
For example:
$x^2 - 4x + 3 = (x-1)(x-3)$ , where $r_1 = 1$ and $r_2 = 3$ are the solutions to the quadratic equation $x^2 - 4x + 3 = 0$ .

Reminder of Log rules

$\log_b(xy) = \log_b(x) + \log_b(y)$
$\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
$\log_b(x^k) = k \log_b(x)$
$\log_{b^k}(x) = \frac{1}{k} \log_b(x)$
$\log_b(b^x) = x$
$\log_b(1) = 0$
$\log_b(b) = 1$
$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

Asymptotes & Intuitive Growth

1. Growth “Ladder”

From slowest to fastest (as $x \to \infty$ ):

Constants: $f(x)=c$ — flat line
Logarithms: $\log x$ — very slow climb
Roots/fractional powers: $x^{1/2},\,x^{1/3}$ — slow but steady
Polynomials: $x^n$ — speed is proportional to degree $n$
Exponentials: $e^x$ or $a^x$ — rapid “explosion”

Visual: log = gentle path; root = shallow hill; $x^2$ = ski slope; $e^x$ = cliff.

2. Horizontal Asymptotes of Rational Functions

For $f(x)=\frac{p(x)}{q(x)},\quad p(x)=a_nx^n+\dots,\;q(x)=b_nx^n+\dots$ both of degree $n$ ,

\lim_{x\to\pm\infty}f(x) =\frac{a_n}{b_n}.

Asymptote:

y=\frac{a_n}{b_n}.

Example 1:
$\lim_{x\to\infty}\frac{3x^4- x+2}{5x^4+7x^2}= \lim_{x\to\infty}\frac{3x^4}{5x^4} = \frac{3}{5}$
so $y=\frac{3}{5}$ is the horizontal asymptote.

Example 2:
$\lim_{x\to\infty}\frac{3x^3 + 2x}{x^5 - 3x^2 + 1}= \lim_{x\to\infty}\frac{3x^3}{x^5} = 0$
so $y=0$ is the horizontal asymptote.

Example 3:
$\lim_{x\to\infty}\frac{x^4 + 2x^2 + 1}{x^3 - 3x^2 + 3x - 1}= \lim_{x\to\infty}\frac{x^4}{x^3} = \infty$
so there is no horizontal asymptote.

Lower‑power terms “vanish” at infinity (e.g. $x^2/x^3=1/x\to0$ ).

Asymptotes come from comparing only the highest‑power terms.

Horizontal asymptotes are found by looking at $x \to \infty$ or $x \to -\infty$ .

If you get $\infty$ or $-\infty$ , there is no horizontal asymptote. Otherwise, the horizontal asymptote is the answer number you get.

3. Vertical & Slant Asymptotes

Vertical at $x=c$ if $\lim_{x\to c^\pm}f(x)=\pm\infty$ .
Slant when $\deg p \geq \deg q+1$ . Divide: $\frac{p(x)}{q(x)}=L(x)+\frac{R(x)}{q(x)},$ then $y=L(x)$ is the slant asymptote since $\lim_{x\to\pm\infty}R(x)/q(x)=0$ .

Example 1:
$\lim_{x\to 1}\frac{1}{(x-1)^2} = \infty$
so $x=1$ is the vertical asymptote.

Example 2:
$\lim_{x\to 0^+}\log_4 (x) = -\infty$
so $x=0$ is the vertical asymptote.

Example 3:
$\lim_{x \to 1}\frac{x^2 - 1}{x-1} = \lim_{x \to 1}\frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1}(x+1) = 2$
so (1, 2) is a hole in the graph.

Euler Number $e$ and its properties

What is $e$ ?

$e$ is a special constant, approximately

e \approx 2.718281828\ldots

It is defined in several equivalent ways. At this point, we will use the limit definition:

Limit definition (compound interest):

e = \lim_{n \to \infty}\Bigl(1 + \tfrac1n\Bigr)^n.

Natural logarithm

The natural logarithm, $\ln(x)$ , is the inverse of the exponential function $e^x$ .

\ln(x) = \log_e(x)

The rest of this set of notes are for your reference and might make more sense later.

Additional properties of e

Properties of $e$

Irrationality: $e$ cannot be expressed as a fraction of two integers.
Transcendence: $e$ is not a root of any nonzero polynomial with integer coefficients.
Limit of a sequence: $e = \lim_{n \to \infty}\Bigl(1 + \tfrac1n\Bigr)^n.$
Infinite series expansion: $e = \sum_{k=0}^\infty \frac{1}{k!}.$
Unique growth rate: among all exponential functions $a^x$ , the function $e^x$ is the only one whose rate of change at $x=0$ is exactly 1.
Derivative and integral: $\frac{d}{dx}e^x = e^x, \qquad \int e^x\,dx = e^x + C.$
Euler’s formula (complex extension): $e^{i\theta} = \cos\theta + i\sin\theta,$ which for $\theta = \pi$ gives the celebrated identity $e^{i\pi} + 1 = 0.$

Why do we care about $e$ ?

Calculus: $e^x$ is its own derivative and integral, making it central in solving differential equations.
Growth and decay models: continuous compound interest, population growth, radioactive decay all use $e$ .
Probability & statistics: appears in the Poisson distribution and normal distribution’s density function.
Complex analysis: Euler’s formula links exponential and trigonometric functions, underpinning Fourier analysis.
Natural logarithm: $\ln(x)$ —the inverse of $e^x$ —simplifies many integrals and solves equations involving products and powers.

Alternative definitions

Series definition:
$e = \sum_{k=0}^\infty \frac{1}{k!} = 1 + 1 + \tfrac12 + \tfrac1{6} + \tfrac1{24} + \cdots.$
Differential equation definition: the unique base for which the function $f(x)=e^x$ satisfies
$\frac{d}{dx}e^x = e^x \quad\text{and}\quad e^0 = 1.$
Via the natural logarithm:
$\ln(e) = 1,\quad\text{and}\quad \ln(x) = \int_1^x \frac{1}{t}\,dt.$
As the unique real number satisfying
$\int_1^e \frac{1}{t}\,dt = 1.$

Applications and examples

Continuous compounding:
If an account pays interest at annual rate $r$ compounded continuously, the balance after $t$ years is $A(t) = A_0\,e^{rt}.$
Differential equations:
The solution to $\frac{dy}{dx} = ky,\quad y(0)=y_0$ is $y(x) = y_0\,e^{kx}.$
Approximating $e$ :
Using the series up to $k=5$ , $e \approx 1 + 1 + \tfrac12 + \tfrac16 + \tfrac1{24} + \tfrac1{120} = 2.716666\ldots$
Compound interest example:
$1000 at 5% interest for 3 years, continuously compounded: $A = 1000\,e^{0.05 \times 3} \approx 1000 \times e^{0.15} \approx 1000 \times 1.1618 = \$1161.83.$

Exercises

1. Limit calculations

Compute the following limits algebraically. (You can use the limit calculator to check your answers and steps.)

a) What is $\lim_{x \to 0} \frac{x^2 + 2x}{x^2 + x}$ ?

b) What is $\lim_{x \to -1} \frac{x^2 - 1}{x+1}$ ?

c) What is $\lim_{x \to 1} \frac{x^4 - 1}{x^2 - 1}$ ?

d) What is $\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$ ?

e) What is $\lim_{x \to 3} \frac{x^2 - 9}{\sqrt{x} - \sqrt{3}}$ ?

f) What is $\lim_{x \to -2} \frac{x^2+x-2}{x^2-4}$ ?

g) What is $\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}$ ?

h) What is $\lim_{x \to 0} \frac{\sqrt{4 + x} - 2}{x}$ ?

2. Limits with infinity

Compute the following limits algebraically. (You can use the limit calculator to check your answers and steps.)

Write 98765 if the limit is $\infty$ and -98765 if the limit is $-\infty$ .

a) What is $\lim_{x \to \infty} \frac{x^3 + 2x^2 + 1}{x^2 + 3}$ ?

b) What is $\lim_{x \to -\infty} \frac{x^3 + 2x^2 + 1}{x^2 + 3}$ ?

c) What is $\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{3x^2 + 2x + 5}$ ?

d) What is $\lim_{x \to \infty} \frac{x^2 + 1}{x^3 + 2x}$ ?

e) What is $\lim_{x \to \infty} \frac{\sqrt{x^2 + 1}}{x + 1}$ ?

f) What is $\lim_{x \to -\infty} \frac{\sqrt{x^2 + 1}}{x + 1}$ ?

g) What is $\lim_{x \to \infty} \frac{\log_3(x)}{\sqrt{x}}$ ?

h) What is $\lim_{x \to \infty} \frac{x^2+2^x}{x^3 + 2x}$ ?

i) What is $\lim_{x \to -\infty} \frac{x^2+2^x}{x^3 + 2x}$ ?

j) What is $\lim_{x \to 0^+} x^{\log_2(x)}$ ?

3. Limit calculations with $e$

Compute the following limits algebraically.

Each of the following limits can be evaluated as $e$ to the power of some expression.

a) What is $x$ : $\lim_{n \to \infty} (1+\frac{1}{2n})^n = e^x$ ?

b) What is $x$ : $\lim_{n \to 0^+} (1+n)^{\frac{3}{n}} = e^x$ ?

c) What is $x$ : $\lim_{n \to \infty} \sqrt{(1+\frac{4}{n})^{n}} = e^x$ ?

d) What is $x$ : $\lim_{n \to \infty} \sqrt[n]{(1+\frac{5}{n})^n} = e^x$ ?

4. Decay exercise

The amount of Carbon-14 in a sample decays according to

N(t) = N_0 e^{-kt}

where $N_0$ is the initial amount of Carbon-14 and $k$ is the decay constant.

a) If the half-life of Carbon-14 is 5730 years, what is the decay constant $k$ ? (compute it in terms of $\ln$ )
b) If a sample has 1000 grams of Carbon-14, how much will be left after 10000 years?
c) If a relic contains 80% of the Carbon-14 it originally had, how old is the relic?

5. Additional general practice

6. Challenge exercises

Try to evaluate the following limits:

a) $\lim_{x \to 0} \frac{e^x - 1}{x}$ (Hint: use the definition of $e$ and the limit laws.)

b) $\lim_{x \to 0} \frac{2^x - 1}{x}$ (Hint: the answer should be expressed in terms of $\ln$ )

c) $\lim_{x \to 0} \frac{b^x - 1}{x}$ for $b > 0$ (Hint: the answer should be expressed in terms of $\ln$ )