Limits, Continuity, and Their Graphical Behavior

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### **1. What is a Limit?**
A **limit** tells us what value a function \( f(x) \) approaches as \( x \) gets closer and closer to a specific number \( a \). It does not matter what \( f(a) \) actually is—only what \( f(x) \) *approaches* as \( x \) nears \( a \).

#### **Key Idea: Approaching vs. Equaling**
- If \( \lim_{x \to a} f(x) = L \), it means \( f(x) \) gets arbitrarily close to \( L \) as \( x \) approaches \( a \).
- The function may or may not be defined at \( x = a \).

#### **Notation**
- \( \lim_{x \to a} f(x) = L \) means "the limit of \( f(x) \) as \( x \) approaches \( a \) is \( L \)."
- We can also write \( f(x) \to L \) as \( x \to a \).

#### **Example 1: Basic Limit**
Find \( \lim_{x \to 2} (3x + 1) \).
**Solution:**
Since \( 3x + 1 \) is a polynomial, it is continuous everywhere. Plug in \( x = 2 \):
\[
\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 7
\]
**Answer:** 7

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### **2. One-Sided Limits**
Sometimes, a function behaves differently when approaching \( a \) from the **left** (\( x \to a^- \)) or the **right** (\( x \to a^+ \)).

#### **Definitions**
- **Left-hand limit (LHL):** \( \lim_{x \to a^-} f(x) = L \) means \( f(x) \) approaches \( L \) as \( x \) approaches \( a \) from values *less than* \( a \).
- **Right-hand limit (RHL):** \( \lim_{x \to a^+} f(x) = L \) means \( f(x) \) approaches \( L \) as \( x \) approaches \( a \) from values *greater than* \( a \).

#### **Example 2: One-Sided Limits**
Find \( \lim_{x \to 1^-} \frac{1}{x-1} \) and \( \lim_{x \to 1^+} \frac{1}{x-1} \).
**Solution:**
- For \( x \to 1^- \), \( x-1 \) is a small negative number (e.g., \( x = 0.999 \), \( x-1 = -0.001 \)). Thus, \( \frac{1}{x-1} \to -\infty \).
- For \( x \to 1^+ \), \( x-1 \) is a small positive number (e.g., \( x = 1.001 \), \( x-1 = 0.001 \)). Thus, \( \frac{1}{x-1} \to +\infty \).

**Answer:**
\[
\lim_{x \to 1^-} \frac{1}{x-1} = -\infty, \quad \lim_{x \to 1^+} \frac{1}{x-1} = +\infty
\]

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### **3. Two-Sided Limits and Existence**
A **two-sided limit** \( \lim_{x \to a} f(x) \) exists **only if** the left-hand and right-hand limits are equal:
\[
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L
\]
If they are not equal, the two-sided limit **does not exist**.

#### **Example 3: Two-Sided Limit**
Does \( \lim_{x \to 0} \frac{|x|}{x} \) exist?
**Solution:**
- For \( x \to 0^- \), \( \frac{|x|}{x} = \frac{-x}{x} = -1 \).
- For \( x \to 0^+ \), \( \frac{|x|}{x} = \frac{x}{x} = 1 \).
Since LHL \( \neq \) RHL, the limit does not exist.

**Answer:** The limit does not exist.

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```figure
{"type":"graph","fns":[{"expr":"abs(x)/x","label":"f(x) = |x|/x"}],"x":[-2,2],"points":[{"x":0,"y":-1,"label":"LHL = -1"},{"x":0,"y":1,"label":"RHL = 1"}],"caption":"LHL ≠ RHL ⇒ two-sided limit does not exist."}

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4. Continuity at a Point

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

1. ( f(a) ) is defined.
2. ( \lim_{x \to a} f(x) ) exists.
3. ( \lim_{x \to a} f(x) = f(a) ).

If any of these fail, ( f(x) ) is discontinuous at ( x = a ).

Types of Discontinuities

Example 4: Checking Continuity

Is ( f(x) = \begin{cases} x^2 & \text{if } x \neq 2 \ 5 & \text{if } x = 2 \end{cases} ) continuous at ( x = 2 )? Solution:

1. ( f(2) = 5 ) (defined).
2. ( \lim_{x \to 2} f(x) = \lim_{x \to 2} x^2 = 4 ).
3. Since ( 4 \neq 5 ), the function is not continuous at ( x = 2 ).

Answer: Discontinuous (removable discontinuity).

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5. Intermediate Value Theorem (IVT)

If ( f(x) ) is continuous on ([a, b]) and ( N ) is any number between ( f(a) ) and ( f(b) ), then there exists a ( c \in (a, b) ) such that ( f(c) = N ).

Example 5: Applying IVT

Show that ( f(x) = x^3 - x - 1 ) has a root in ([1, 2]). Solution:

1. Check continuity: ( f(x) ) is a polynomial ⇒ continuous everywhere.
2. Evaluate at endpoints:
   • ( f(1) = 1 - 1 - 1 = -1 ).
   • ( f(2) = 8 - 2 - 1 = 5 ).
3. Since ( 0 ) is between ( -1 ) and ( 5 ), by IVT, there exists ( c \in (1, 2) ) such that ( f(c) = 0 ).

Answer: Root exists in (1, 2).

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6. Limits and Continuity in Graphs

Example 6: Identifying Discontinuities

Sketch the graph of ( f(x) = \begin{cases} x + 1 & \text{if } x < 0 \ x^2 & \text{if } x \geq 0 \end{cases} ) and identify discontinuities. Solution:

1. For ( x < 0 ): straight line ( y = x + 1 ).
2. For ( x \geq 0 ): parabola ( y = x^2 ).
3. At ( x = 0 ):
   • LHL: ( \lim_{x \to 0^-} (x + 1) = 1 ).
   • RHL: ( \lim_{x \to 0^+} x^2 = 0 ).
   • Since LHL ( \neq ) RHL, there is a jump discontinuity at ( x = 0 ).

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Exam Tip

1. Always check both sides for limits at points where the function changes definition (e.g., piecewise functions).
2. Three conditions for continuity: defined, limit exists, limit equals the function value. If any fail, it’s discontinuous.
3. IVT is your friend: Use it to prove roots exist in an interval if the function is continuous and changes sign.
4. Graphs are clues: Sketch the function to visualize limits and discontinuities.
5. Common mistakes:
   • Assuming ${\lim }_{x\to a}f(x)=f(a)$ (it’s only true if continuous).
   • Forgetting to check both sides for one-sided limits.
   • Misidentifying discontinuity types (e.g., calling a jump discontinuity "removable").

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NEB Board-Style Questions

Short Answer (5 marks each)

1. Find ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$.
2. Determine if $f(x)=\{\begin{array}{ll}\frac{\sin x}{x}& \text{if }x\ne 0\\ 1& \text{if }x=0\end{array}$ is continuous at $x=0$.
3. Sketch the graph of $f(x)=\frac{1}{x+2}$ and identify any discontinuities.
4. Use the Intermediate Value Theorem to show that $f(x)=x^3+x-3$ has a root in $[1,2]$.
5. Find ${\lim }_{x\to -2^{+}}\frac{x+2}{x^2-4}$ and ${\lim }_{x\to -2^{-}}\frac{x+2}{x^2-4}$. Does the two-sided limit exist?

Long Answer (10 marks)

6. Let $f(x)=\{\begin{array}{ll}{x}^2+1& \text{if }x\le 1\\ 2x& \text{if }x>1\end{array}$.
   • Find ${\lim }_{x\to 1^{-}}f(x)$ and ${\lim }_{x\to 1^{+}}f(x)$.
   • Is $f(x)$ continuous at $x=1$? Justify.
   • Sketch the graph of $f(x)$ and label all key points.
7. Explain the difference between a removable discontinuity and a jump discontinuity with examples. How would you identify each from a graph?

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Note: Practice sketching graphs of piecewise functions and rational functions (e.g., $\frac{1}{x}$, $\frac{1}{x-1}$) to recognize discontinuities intuitively. The NEB exam often tests graphical interpretation!