lecture_EE1G1_lec2_math_limits_2025-09-09

Lecture 2: Introduction to Limits

📝 Summary

This lecture introduces the fundamental concept of a limit in calculus. A limit describes the value that a function approaches as the input approaches some value. Understanding limits is crucial for defining continuity, derivatives, and integrals.

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💡 Key Concepts

1. Intuitive Definition of a Limit

The limit of a function f(x) as x approaches a number c is a number L, if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to c, but not equal to c.

• Notation: lim f(x) = L as x -> c or ${\lim }_{x\to c}f(x)=L$

2. One-Sided Limits

• Right-Hand Limit: The value the function approaches as x approaches c from the right side (values greater than c).
  • Notation: ${\lim }_{x\to c^{+}}f(x)$
• Left-Hand Limit: The value the function approaches as x approaches c from the left side (values less than c).
  • Notation: ${\lim }_{x\to c^{-}}f(x)$

For a limit to exist at c, the left-hand limit must equal the right-hand limit. ${\lim }_{x\to c}f(x)=L⟺{\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)=L$

3. Limits at Infinity

This describes the behavior of a function as x grows infinitely large (positive or negative). This is related to horizontal asymptotes.

• Notation: ${\lim }_{x\to \infty }f(x)=L$ or ${\lim }_{x\to -\infty }f(x)=L$

4. Indeterminate Forms

Sometimes, direct substitution leads to forms like 0/0 or ∞/∞. These are indeterminate forms, and they require special techniques to solve.

• Common Techniques:
  • Factoring and canceling
  • Multiplying by the conjugate
  • L’Hôpital’s Rule

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🧮 Quick Examples

Example 1: A Simple Limit

• Problem: Find ${\lim }_{x\to 2}(3x^2-2x+1)$
• Solution: Since this is a polynomial, we can use direct substitution.
  • $3(2{)}^2-2(2)+1=3(4)-4+1=12-4+1=9$

Example 2: Factoring and Canceling (Indeterminate Form 0/0)

• Problem: Find ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$
• Solution: Direct substitution gives (9-9)/(3-3) = 0/0. We need to simplify.
  • Factor the numerator: ${\lim }_{x\to 3}\frac{(x-3)(x+3)}{x-3}$
  • Cancel the (x-3) term: ${\lim }_{x\to 3}(x+3)$
  • Now substitute x=3: $3+3=6$

Example 3: L’Hôpital’s Rule (Indeterminate Form 0/0)

• Problem: Find ${\lim }_{x\to 0}\frac{\sin (x)}{x}$
• Solution: This is a famous limit that equals 1. It’s an indeterminate form 0/0. We can use L’Hôpital’s Rule by taking the derivative of the top and bottom.
  • Derivative of sin(x) is cos(x).
  • Derivative of x is 1.
  • ${\lim }_{x\to 0}\frac{\cos (x)}{1}=\frac{\cos (0)}{1}=\frac{1}{1}=1$

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📚 Related Topics

• LHopitals-Rule

✍️ Notes

One-Sided Limits

One-sided limits describe the behavior of a function as its input approaches a certain value from either the left or the right side. They are crucial for determining if a general limit exists at a specific point.

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💡 Explanation

Left-Hand Limit

The left-hand limit is the value that the function f(x) approaches as x approaches c from values less than c.

• Notation: ${\lim }_{x\to c^{-}}f(x)=L$

Right-Hand Limit

The right-hand limit is the value that the function f(x) approaches as x approaches c from values greater than c.

• Notation: ${\lim }_{x\to c^{+}}f(x)=R$

For a general limit to exist at x=c, the left-hand limit must equal the right-hand limit (L must equal R). If they are different, the limit at c does not exist.

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🖼️ Diagrams & Visuals

This diagram shows a function where the left-hand limit as x approaches c is L, and the right-hand limit is R. Since $L\ne R$, the overall limit of $f(x)$ as $x\to c$ does not exist.

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📚 Related Resources

• Limit-of-a-function
• Continuity

Limit of a function

The limit of a function describes the value that a function approaches as the input (or index) approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, and integrals.

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💡 Explanation

The concept of a limit is about approaching a value, not necessarily reaching it. We write the limit of a function f(x) as x approaches c as L:

${\lim }_{x\to c}f(x)=L$

This means that as x gets closer and closer to c (from both sides), the value of f(x) gets closer and closer to L. The function does not need to be defined at x=c for the limit to exist.

For a limit to exist, the function must approach the same value from both the left and the right.

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🖼️ Diagrams & Visuals

This diagram illustrates that as x approaches a point c from both the left and the right, the function f(x) approaches the value L. The open circle at (c, L) signifies that the function’s value at c itself doesn’t affect the limit.

\begin{document}
\begin{tikzpicture}
    \begin{axis}[
        axis x line=bottom,
        axis y line=left,
        xlabel={$x$},
        ylabel={$f(x)$},
        xmin=-1, xmax=5,
        ymin=-1, ymax=4,
        xtick={2},
        ytick={3},
        xticklabels={$c$},
        yticklabels={$L$},
        grid=both,
        grid style={dashed, gray!30},
        width=10cm,
        height=8cm,
    ]
    % Plot a sample function
    \addplot[domain=0:4, samples=100, blue, thick] {-(x-2)^2 + 3};
    
    % Dashed lines to show the limit
    \draw[dashed] (2, -1) -- (2, 3);
    \draw[dashed] (-1, 3) -- (2, 3);
    
    % Open circle at the limit point
    \fill[white] (2, 3) circle (2.5pt);
    \draw[blue, thick] (2, 3) circle (2.5pt);
    
    % Arrows showing the approach
    \node[pin=135:{$x \to c^-$}] at (1.5, 2.75) {};
    \draw[->, red, thick] (1, 2) -- (1.8, 2.95);
    
    \node[pin=45:{$x \to c^+$}] at (2.5, 2.75) {};
    \draw[->, red, thick] (3, 2) -- (2.2, 2.95);
    
    \node[pin={[pin edge={->,red,thick}]left:{$f(x) \to L$}}] at (1.9,3.5) {};
 
    \end{axis}
\end{tikzpicture}
\end{document}

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📚 Related Resources

• Continuity
• Derivatives

✍️ Notes on Indeterminate Forms

Indeterminate forms are expressions that arise in the context of limits, where the value of the limit cannot be determined by simple substitution. The most common examples are 0/0 and ∞/∞. These forms do not have a defined value and require further algebraic manipulation or other techniques to evaluate the limit.

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💡 Explanation

When evaluating a limit ${\lim }_{x\to c}f(x)$, if direct substitution of c into the function results in one of the following forms, the limit is said to be “indeterminate”:

• $\frac{0}{0}$
• $\frac{\infty }{\infty }$
• $0\cdot \infty$
• $\infty -\infty$
• $1^{\infty }$
• $0^0$
• ${\infty }^0$

An indeterminate form does not mean the limit does not exist. It simply means that the initial method of substitution is insufficient.

Common Techniques for Resolution

1. Algebraic Manipulation:
   • Factoring and Canceling: Useful for rational functions.
   • Multiplying by the Conjugate: Useful for limits involving square roots.
2. L’Hôpital’s Rule: If the limit is of the form 0/0 or ∞/∞, we can take the derivative of the numerator and the denominator separately and then evaluate the limit.

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🖼️ Diagrams & Visuals

This flowchart shows the process for handling limits. If direct substitution results in an indeterminate form, you must apply other techniques like algebraic manipulation or L’Hôpital’s Rule to find the true value of the limit.

\begin{document}
\begin{tikzpicture}[
    node distance=2cm,
    auto,
    block/.style={rectangle, draw, fill=blue!20, text width=8em, text centered, rounded corners, minimum height=3em},
    decision/.style={diamond, draw, fill=green!20, text width=6em, text centered, minimum height=3em},
    line/.style={draw, -latex'}
]
    % Nodes
    \node [block] (start) {Evaluate $\lim_{x \to c} f(x)$ by substituting $x=c$};
    \node [decision, below of=start, node distance=3cm] (check) {Result is an indeterminate form? (e.g., 0/0)};
    \node [block, below of=check, node distance=3cm] (resolve) {Apply techniques: \begin{itemize} \item Factoring \item Conjugate \item L'Hôpital's Rule \end{itemize}};
    \node [block, right of=check, node distance=4cm] (defined) {Limit is the resulting value};
    \node [block, below of=resolve] (final) {Evaluate the new limit};
 
    % Edges
    \path [line] (start) -- (check);
    \path [line] (check) -- node [near start] {Yes} (resolve);
    \path [line] (check) -- node [near start] {No} (defined);
    \path [line] (resolve) -- (final);
    \path [line] (final) |- (defined);
\end{tikzpicture}
\end{document}

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📚 Related Resources

• LHopitals-Rule

✍️ Notes on Limits at Infinity

Limits at infinity are used to describe the behavior of a function as the input x grows without bound, either in the positive or negative direction. This concept is directly related to horizontal asymptotes of a function’s graph.

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💡 Explanation

We are interested in what value f(x) approaches as x becomes very large.

• Limit as x approaches positive infinity: ${\lim }_{x\to \infty }f(x)=L$ This means that f(x) gets arbitrarily close to L as x increases without bound. The line y=L is a horizontal asymptote.

• Limit as x approaches negative infinity: ${\lim }_{x\to -\infty }f(x)=M$ This means that f(x) gets arbitrarily close to M as x decreases without bound. The line y=M is a horizontal asymptote.

A common technique for rational functions is to divide every term in the numerator and denominator by the highest power of x in the denominator.

Integration by parts

$\int f(x)g(x)dx=g(x)F(x)-\int g^{\prime }(x)F(x)dx$

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🖼️ Diagrams & Visuals

This diagram shows a function with a horizontal asymptote at y=L. As x approaches both positive and negative infinity, the function f(x) gets closer and closer to the value L.

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📚 Related Resources

• Limit-of-a-function
• Asymptotes