Limits

Formal Definition

ε-δ Definition

For a function f(x) and a point a, we say:

\[\lim_{x \to a} f(x) = L\]

if for every ε > 0, there exists a δ > 0 such that:

if 0 <

x - a

< δ, then

f(x) - L

< ε.

One-Sided Limits

Right-hand limit: $\lim_{x \to a^+} f(x) = L$ if for every ε > 0, there exists a δ > 0 such that if a < x < a + δ, then

f(x) - L

< ε.

Left-hand limit: $\lim_{x \to a^-} f(x) = L$ if for every ε > 0, there exists a δ > 0 such that if a - δ < x < a, then

f(x) - L

< ε.

A limit exists if and only if both one-sided limits exist and are equal: $\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

Limit Laws

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

Sum: $\lim_{x \to a} [f(x) + g(x)] = L + M$
Difference: $\lim_{x \to a} [f(x) - g(x)] = L - M$
Product: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$
Constant Multiple: $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$
Power: $\lim_{x \to a} [f(x)]^n = L^n$, where n is a positive integer
Root: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}$, if L > 0 for even n
Composition: If $\lim_{x \to a} g(x) = M$ and $\lim_{y \to M} h(y) = N$, then $\lim_{x \to a} h(g(x)) = N$

Special Limits

Trigonometric Limits:
- $\lim_{x \to 0} \frac{\sin x}{x} = 1$
- $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
- $\lim_{x \to 0} \frac{\tan x}{x} = 1$
Exponential Limits:
- $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$
- $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
- $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$
Algebraic Limits:
- $\lim_{x \to \infty} \frac{x^n}{e^x} = 0$ for any positive n
- $\lim_{x \to \infty} \frac{\ln x}{x} = 0$
- $\lim_{x \to \infty} \frac{a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0}{b_mx^m + b_{m-1}x^{m-1} + \cdots + b_0} = \begin{cases} \frac{a_n}{b_m} & \text{if } n = m \ 0 & \text{if } n < m \ \infty & \text{if } n > m \end{cases}$

Indeterminate Forms

$\frac{0}{0}$: When both numerator and denominator approach 0
$\frac{\infty}{\infty}$: When both numerator and denominator approach infinity
$0 \cdot \infty$: Product of a quantity approaching 0 and another approaching infinity
$\infty - \infty$: Difference of two quantities approaching infinity
$0^0$, $1^\infty$, $\infty^0$: Powers with special limiting behaviors

Techniques for Evaluating Limits

Direct Substitution

If f(x) is continuous at x = a, then $\lim_{x \to a} f(x) = f(a)$

Algebraic Manipulation

Factoring
Rationalization
Using conjugates
Multiplying by a clever form of 1

Using Standard Limits

Apply known limits like $\lim_{x \to 0} \frac{\sin x}{x} = 1$

L’Hôpital’s Rule

For indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

Taylor Series Expansion

For complex limits, expand functions using Taylor series: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$

Solving Limit Problems

Example 1: Basic Substitution

$\lim_{x \to 3} (x^2 + 2x - 1)$ Direct substitution: $3^2 + 2(3) - 1 = 9 + 6 - 1 = 14$

Example 2: Factoring Approach

$\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$ Factor: $\frac{(x+4)(x-4)}{x-4} = x+4$ for $x \neq 4$ Therefore: $\lim_{x \to 4} \frac{x^2 - 16}{x - 4} = 4 + 4 = 8$

Example 3: Trigonometric Limit

$\lim_{x \to 0} \frac{\sin(3x)}{2x}$ Rewrite: $\frac{\sin(3x)}{2x} = \frac{3}{2} \cdot \frac{\sin(3x)}{3x}$ Apply the standard limit: $\lim_{x \to 0} \frac{\sin(3x)}{2x} = \frac{3}{2} \cdot 1 = \frac{3}{2}$

Example 4: Complex Fraction

$\lim_{x \to 0} \frac{\sin x - x}{x^3}$ Use Taylor series: $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ Substituting: $\frac{\sin x - x}{x^3} = \frac{-\frac{x^3}{6} + \frac{x^5}{120} - \cdots}{x^3} = -\frac{1}{6} + \frac{x^2}{120} - \cdots$ Therefore: $\lim_{x \to 0} \frac{\sin x - x}{x^3} = -\frac{1}{6}$

Example 5: Exponential Form

$\lim_{x \to 0^+} x^{\text{sign}x}$ Since $x \to 0^+$, sign(x) = 1 Therefore: $\lim_{x \to 0^+} x^{\text{sign}x} = \lim_{x \to 0^+} x^1 = 0$

Strategies for Difficult Limits

Identify the form (direct substitution, indeterminate)
Choose appropriate technique based on the form
Transform the expression to a more manageable form
Apply limit laws and standard limits
Check results for reasonableness

For trigonometric functions:

Convert to sin/cos when dealing with tan, cot, sec, csc
Use trigonometric identities to simplify
Apply standard limits for common forms

For rational functions:

Factor when possible
Divide by highest power for limits at infinity
Look for opportunities to cancel terms

For exponential and logarithmic functions:

Use properties of logarithms to simplify
Apply the relationship between exponentials and logarithms
Consider Taylor series for complex expressions

See:

Limit Exercises

1. Basic Limits

Compute the limit $\lim_{x \to 0} \frac{\sin(x^2) - x^2}{x^3}$
Compute the limit $\lim_{x \to 0} \frac{\ln(x^2 + 1)}{\cos x - 1 + \frac{1}{2}x^2}$
Compute the limit $\lim_{x \to 0}\frac{1-e^{2x^2}+ 2\ln(1-x^2)}{x^4}$
Verify by means of the definition that: a) $\lim_{x \to 0^+} \frac{1}{x^3-1} = -1$ b) $\lim_{x \to -1} \frac{1}{1+x^2} = \frac{1}{2}$
Compute, if it exists, the limit $\lim_{x \to 0} x^{\text{sign}x}$
Given the function $f(x) = \begin{cases} x+2 & x < 0
e^x & x \in [0,1]
2-x & x > 1 \end{cases}$

Compute using the definition: $\lim_{x \to 0^-}f(x)$, $\lim_{x \to 0^+}f(x)$, $\lim_{x \to 1^-}f(x)$, $\lim_{x \to 1^+}f(x)$
Prove that the limit $\lim_{x \to 0}\frac{ x (2-x^2+x)}{x}$ does not exist.

2. Standard Form Limits

Compute the following limits: a) $\lim_{x \to 0}\frac{1+x^2}{1-x}$ b) $\lim_{x \to 0}\sqrt{2+3x}$ c) $\lim_{x \to 0}x\sin x$ d) $\lim_{x \to 1}\frac{x+1}{x^2-2x+1}$ e) $\lim_{x \to 4}(x\sqrt{x}+1)$ f) $\lim_{x \to 4}\frac{x^2-16}{x-4}$ g) $\lim_{x \to 5}\frac{\sqrt{x-1}-2}{x-5}$ h) $\lim_{x \to 0}\frac{x^3-3x^2+4x}{x^5-x}$ i) $\lim_{x \to 5}\frac{x-5}{\sqrt{x}-\sqrt{5}}$ j) $\lim_{x \to 0}\frac{x+2\sqrt{x}+3x^3}{5\sqrt{x}+x^2}$ k) $\lim_{x \to 0}\frac{x^3-1}{1-x^2}$ l) $\lim_{x \to 0}\frac{\sqrt{x+3}-\sqrt{3}}{x}$ m) $\lim_{x \to -\frac{1}{2}}\frac{2x+1}{\sqrt{8x^2-1}-1}$

3. Exponential and Infinite Limits

Compute the following limits: a) $\lim_{x \to 0^+}\frac{1+2^{\frac{1}{x}}}{3+2^{\frac{1}{x}}}$ b) $\lim_{x \to 0^-}\frac{1+2^{\frac{1}{x}}}{3+2^{\frac{1}{x}}}$ c) $\lim_{x \to +\infty}\frac{x^2+3}{4x^2+x}$ d) $\lim_{x \to +\infty}\frac{3-x^2+3x\sqrt{x}}{4x^2\sqrt{x}+x}$ e) $\lim_{x \to +\infty}\frac{x^2+\sin x}{1+x+3x^2}$ f) $\lim_{x \to +\infty}\frac{\sqrt{1+x^2}+\sqrt{x}}{\sqrt{x}-x}$ g) $\lim_{x \to -\infty}\frac{\lg(3+\sin x)}{x}$ h) $\lim_{n \to +\infty}\left(1+\frac{1}{n}\right)^{2n}$ i) $\lim_{n \to +\infty}\left(\frac{3n+2}{3n+1}\right)^n$ j) $\lim_{x \to 0}\frac{3^{\sin x}-1}{x}$

4. Trigonometric Limits

Compute the following limits using the standard limits: a) $\lim_{x \to 0}\frac{\tan x}{x}$ b) $\lim_{x \to 0}\frac{\sin(3x)}{2x}$ c) $\lim_{x \to 0}\frac{\sin x + x}{\tan x + x}$ d) $\lim_{x \to 0}\frac{1}{1-\cos x}$ e) $\lim_{x \to 0}\frac{\sin(x^2)}{1-\cos x}$ f) $\lim_{x \to 0}\frac{1-\cos(3x)}{(\sin x)^2}$ g) $\lim_{x \to 0}\frac{\sin(\sin x)}{x}$ h) $\lim_{x \to 0}\frac{\sin(\cos x)}{x}$ i) $\lim_{x \to 0}\frac{\sin x+3x}{x^2+\cos x}$ j) $\lim_{x \to 0^+}\frac{\sin x}{x+\sqrt{x}}$ k) $\lim_{x \to 0}\frac{\sin(x^2)}{x\lg(1+x)}$

5. Sequence Limits

Compute the following limits: a) $\lim_{n \to \infty} \left(\frac{2n-1}{n+1}\right)^{n-1}$ b) $\lim_{n \to \infty} \sqrt{\frac{n^2+2}{2n^2-1}}$ c) $\lim_{x \to \infty} \sqrt{x^2+1}-x$ d) $\lim_{n \to \infty} \frac{n\sin n}{n^2-1}$ e) $\lim_{n \to +\infty} n^2\left(\cos\left(\frac{1}{n}\right)-1\right)$

6. Complex Limits

Compute the following limits: a) $\lim_{x \to 0} \frac{e^x-1-x}{1-\cos x - \frac{1}{2}x^2}$ b) $\lim_{x \to 0} \frac{e^x-1-x}{1-\cos x}$ c) $\lim_{x \to 0} \frac{\sin x - x}{1-\cos(x^{3/2})}$ d) $\lim_{x \to 0} \frac{1-\cos(x^3)}{x^2\sqrt{1+x}-1}$ e) $\lim_{x \to 0} \frac{\sqrt{x+3}-\sqrt{3}}{1-\sqrt{x+1}}$ f) $\lim_{x \to -\infty} \frac{x}{\sqrt{1-x+3|x|}}$