Session 05-01 - Limits & Continuity Through Graphs

Section 05: Differential Calculus

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Review from Section 04

Test your understanding from Advanced Functions

  1. What happens to \(f(x) = \frac{1}{x-2}\) as \(x\) approaches 2 from the right?

  2. For the rational function \(g(x) = \frac{x^2-4}{x-2}\), does the graph have a hole or a vertical asymptote at \(x = 2\)?

  3. What is the horizontal asymptote of \(f(x) = \frac{3x^2 + 1}{x^2 - 4}\)?

Homework Discussion

Solution from Tasks 04-04

15 minutes - presentation and discussion

  • Present difficult problems
  • Discuss your potential problems
  • Share strategies for your solutions

Your questions from Section 04

Focus on rational functions and asymptotic behavior

  • Challenges with finding asymptotes of rational functions
  • Interpreting end behavior and horizontal asymptotes
  • Understanding vertical asymptotes and domain restrictions
  • Questions about transformations of functions

Today we formalize the “limit” concept that explains all this asymptotic behavior!

Learning Objectives

What You’ll Master Today

  • Understand limits intuitively through graphical analysis
  • Evaluate one-sided limits and determine when limits exist
  • Identify types of discontinuities in real-world functions
  • Apply continuity concepts to business scenarios
  • Connect abstract math to practical decision-making
  • Build the foundation for derivatives in the next session

Limits are the foundation of calculus!

The Limit Concept

From Asymptotes to Limits

Remember rational functions from Section 04?

  • You saw that \(f(x) = \frac{1}{x}\) gets closer to 0 as \(x\) gets larger
  • You identified vertical asymptotes where functions “blow up”
  • You found horizontal asymptotes showing long-term behavior

Today’s Concept: A limit describes what value a function approaches as the input approaches a specific value.

Notation: \(\lim_{x \to a} f(x) = L\)

“The limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\)”

Visual Understanding of Limits

We write: \(\lim_{x \to 2} f(x) = 4\)

Graph of f(x) = (x² − 4)/(x − 2): a straight line with a hole at (2, 4). Arrows along the line from the left and from the right both point at the hole, so the values approach 4 as x approaches 2.

Look at the Following

Price per unit against quantity: a constant 10 EUR below 100 units, then a constant 8 EUR bulk price from 100 units on. An open circle at 10 EUR and a filled one at 8 EUR mark the jump discontinuity at 100 units.

Question: What happens to the price as quantity approaches 100?

One-Sided Limits

Approaching from Two Directions

Sometimes the approach direction matters!

  • Left-hand limit: \(\lim_{x \to a^-} f(x)\) (approaching from the left)
  • Right-hand limit: \(\lim_{x \to a^+} f(x)\) (approaching from the right)
  • Two-sided limit exists when: Left limit = Right limit

One-sided limits (like \(\lim_{x \to a^-} f(x)\)) always describe approaching from one direction!

Example: Tax Brackets

This is how taxes are often handled:

Step function of the tax rate against taxable income: 0% below 10,000 EUR, 14% from 10,000 EUR, 24% from 30,000 EUR and 42% from 60,000 EUR. The rate jumps at every bracket boundary.

Marginal tax rates, not the effective rate on total income.

Evaluating One-Sided Limits

For the tax function at 30,000 EUR:

  • From the left: \(\lim_{x \to 30000^-} T(x) = 14\%\)
    • Income of 29,999 EUR → 14% rate
    • Income of 29,999.99 EUR → 14% rate
  • From the right: \(\lim_{x \to 30000^+} T(x) = 24\%\)
    • Income of 30,001 EUR → 24% rate
    • Income of 30,000.01 EUR → 24% rate
  • Two-Sided limit doesn’t exist at 30,000 EUR (jump discontinuity)

When Limits Exist

Criterion for Two-Sided Limits

A two-sided limit \(\lim_{x \to a} f(x)\) exists if and only if:

  1. The left-hand limit exists: \(\lim_{x \to a^-} f(x) = L_1\)
  2. The right-hand limit exists: \(\lim_{x \to a^+} f(x) = L_2\)
  3. They are equal: \(L_1 = L_2\)

Two-sided limits (written simply as \(\lim_{x \to a} f(x)\)) require agreement from both sides. When we say “the limit doesn’t exist,” we typically mean the two-sided limit.

The limit describes approaching behavior, not the actual value at the point.

Types: Limit Exists

Graph of f(x) = (x² − 1)/(x − 1): a straight line with an open circle, a hole, at (1, 2). The function is undefined at x = 1, yet from both sides the values approach 2.

\(\lim_{x \to 1} f(x) = 2\) (even though \(f(1)\) is undefined)

Types: Jump Discontinuity

Two rising line pieces that do not meet at x = 1: the left piece ends at an open circle at height 2, the right piece starts at a filled point at height 3. The one-sided limits 2 and 3 differ.

Left limit ≠ Right limit, so limit doesn’t exist (two-sided!)

Types: Infinite Discontinuity

Graph of 1/x with a dashed vertical asymptote at x = 0: the left branch plunges towards minus infinity and the right branch shoots up towards plus infinity as x approaches 0.

\(\lim_{x \to 0^-} f(x) = -\infty\) and \(\lim_{x \to 0^+} f(x) = +\infty\)

Quick Practice

Individual Exercise

Work individually, then compare

Piecewise graph for the exercise: the parabola x² for x < 1 ends at an open circle (1, 1); a horizontal segment at height 2 runs from a filled point (1, 2) to (3, 2); a line falls from (3, 2) to an open circle at (5, 0); from there a line rises gently to the right.

Your Tasks

Find the following:

  1. \(\lim_{x \to 1^-} f(x)\) and \(\lim_{x \to 1^+} f(x)\)
  2. \(\lim_{x \to 3} f(x)\)
  3. \(\lim_{x \to 5^-} f(x)\) and \(\lim_{x \to 5^+} f(x)\)

Trace the curve from each direction. The y-value you approach is the limit.

Break

Continuity at a Point

The Three Conditions for Continuity

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

  1. \(f(a)\) is defined (the function has a value at \(a\))
  2. \(\lim_{x \to a} f(x)\) exists (the limit exists)
  3. \(\lim_{x \to a} f(x) = f(a)\) (limit equals the function value)
NoteBusiness Interpretation

Continuity means “no sudden jumps”, important for:

  • Smooth production processes
  • Gradual price changes
  • Predictable cost functions

Visual Continuity Test

Three panels compared at x = 0. Left: parabola x² with f(0) = 0 equal to the limit, continuous. Middle: the same parabola with a hole at (0, 0) and the function value moved up to (0, 1), a removable discontinuity. Right: two line pieces approaching 1 from the left and −1 from the right, a jump discontinuity.

Example: Production Capacity

A factory’s cost function changes at 1000 units:

\[C(x) = \begin{cases} 5x + 1000 & \text{if } x < 1000 \\ 3x + 3000 & \text{if } x \geq 1000 \end{cases}\]

Question: Any idea how this might look?

Production Capacity: Visualization

Total cost against production quantity in two linear segments: a steeper one below 1,000 units and a flatter one from 1,000 units on. Both reach 6,000 EUR at 1,000 units, so the cost function has a kink but no jump.

This is continuous! The company planned the transition carefully.

Guided Practice

Practice Set A: Limit Evaluation

Four graphs for limit practice. Function 1: a rising straight line with a hole at (1, 2). Function 2: a parabola x² up to x = 2 that continues as the line 2x, joined at the filled point (2, 4). Function 3: a V shape with its corner at (1, 0). Function 4: a staircase at heights 1, 2, 3 and 4 that jumps up at x = 1, 2 and 3.

Practice Set A: Questions

Work individually for 5 minutes

Evaluate the limits from the previous graphs:

  1. Function 1: \(\lim_{x \to 1} f(x)\)
  2. Function 2: \(\lim_{x \to 2^-} f(x)\) and \(\lim_{x \to 2^+} f(x)\)
  3. Function 3: \(\lim_{x \to 1} f(x)\)
  4. Function 4: Does \(\lim_{x \to 2} f(x)\) exist?

Practice Set B: Continuity Analysis

For the function below, determine where it is continuous:

\[f(x) = \begin{cases} x^2 + 1 & \text{if } x < 0 \\ 2 & \text{if } x = 0 \\ -x + 3 & \text{if } 0 < x < 3 \\ \frac{6}{x-3} & \text{if } x > 3 \end{cases}\]

  1. Check continuity at \(x = 0\)
  2. Check continuity at \(x = 3\)
  3. Sketch the function
  4. Where is \(f\) continuous?

Coffee Break

Business Applications

Shipping Cost Models

Online retailers often use step functions for shipping:

Step function of shipping cost against order value: 9.99 EUR below 25 EUR, 5.99 EUR from 25 EUR, 2.99 EUR from 50 EUR and free from 100 EUR. The cost jumps down at each threshold.

Question: How does this pricing structure influence customer behavior?

Government Price Intervention

Demand exceeds threshold, government releases emergency reserves:

Price falls linearly with quantity demanded, from 80 EUR to 50 EUR at the intervention threshold of 60. There it drops instantly by 30 EUR to 20 EUR and keeps falling: a jump discontinuity.

Real-World

Examples of Discontinuous Price Functions

  • Government reserves released when prices exceed threshold
  • Rent control kicking in above certain income levels
  • Tiered utility pricing with usage thresholds
  • Tariffs applied when imports exceed quotas

Policy interventions often create discontinuities. Limits help us analyze prices “just before” and “just after” the threshold.

Economic: Long-Run Average Cost

Business Context: A company’s total cost function is \(C(x) = 5000 + 20x + 0.001x^2\).

Question: What happens to average cost in the long run?

\[\lim_{x \to \infty} \overline{C}(x) = \lim_{x \to \infty} \left(\frac{5000}{x} + 20 + 0.001x\right)\]

  • \(\frac{5000}{x} \to 0\) (fixed costs spread over many units)
  • \(20 \to 20\) (constant variable cost)
  • \(0.001x \to \infty\) (diseconomies of scale eventually dominate)

More Limit Practice

Mastering Limit Evaluation

Evaluating limits requires recognizing the type of limit!

  1. Direct substitution first - if the function is continuous, plug in!
  2. Form 0/0 - factor and simplify
  3. Form ∞/∞ - compare degrees (rational functions)
  4. One-sided limits differ - limit doesn’t exist
  5. Infinite limits - vertical asymptote behavior

Forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) are indeterminate - the form alone doesn’t tell you the answer. The limit could be any number! You must apply techniques (factoring, simplifying) to find the actual value.

Practice Set A: Algebraic Limits

Work individually for 8 minutes

Evaluate these limits:

  1. \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}\)

  2. \(\displaystyle\lim_{x \to 0} \frac{x^3 - x}{x}\)

  3. \(\displaystyle\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}\)

Practice Set B: Limits at Infinity

Work individually for 5 minutes

Evaluate these limits:

  1. \(\displaystyle\lim_{x \to \infty} \frac{3x^2 + 2x}{5x^2 - 1}\)

  2. \(\displaystyle\lim_{x \to \infty} \frac{x + 1}{x^3 - 2}\)

  3. \(\displaystyle\lim_{x \to -\infty} \frac{2x^3}{x^2 + 1}\)

  4. \(\displaystyle\lim_{x \to \infty} \frac{4x^2 - 3x + 1}{2x^2 + x}\)

Practice Set C: One-Sided & Infinite

Work individually for 5 minutes

  1. \(\displaystyle\lim_{x \to 0^+} \frac{1}{x}\)

  2. \(\displaystyle\lim_{x \to 0^-} \frac{1}{x}\)

  3. Does \(\displaystyle\lim_{x \to 0} \frac{1}{x}\) exist? Why or why not?

  4. \(\displaystyle\lim_{x \to 2^-} \frac{x}{x - 2}\) and \(\displaystyle\lim_{x \to 2^+} \frac{x}{x - 2}\)

Team Problem-Solving

Challenge: Production Planning

A manufacturing company has this cost structure:

  • Small batches (< 500 units): \(C_1(x) = 20x + 500\)
  • Medium batches (500-2000 units): \(C_2(x) = 15x + 750\)
  • Large batches (> 2000 units): \(C_3(x) = 12x + 1500\)

Challenge: Tasks

Work in groups of 3-4

  1. Write the complete piecewise cost function
  2. Identify all points of discontinuity
  3. Calculate limits at each transition point
  4. Determine which transitions are continuous
  5. Graph the cost function
  6. Recommend: Should the company smooth these transitions?

Wrap-Up & Takeaways

Today’s Essential Concepts

  • Limits describe approaching behavior - what happens as we get close
  • One-sided limits help analyze jumps and breaks
  • Continuity requires three conditions - defined, limit exists, they match
  • Discontinuity types: removable (hole), jump, infinite (asymptote)
  • Business functions often have discontinuities - and that’s okay!
  • Graphical analysis is powerful for understanding limits

Limits are the foundation of calculus. Next session, we’ll use them to define derivatives!

Final Assessment

Quick Check

Work individually, then we compare

  1. For \(f(x) = \frac{x^2 - 9}{x - 3}\): Find \(\lim_{x \to 3} f(x)\) and is \(f\) continuous at \(x = 3\)?

  2. A parking fee function: \[P(t) = \begin{cases} 5 & \text{if } 0 < t \leq 2 \\ 10 & \text{if } 2 < t \leq 4 \\ 20 & \text{if } t > 4 \end{cases}\] Where is \(P(t)\) discontinuous?

Next Session Preview

Coming Up: The Derivative

  • From average rate of change to instantaneous rate
  • The derivative as a limit
  • Marginal cost, revenue, and profit
  • Finding tangent lines to curves

Complete Tasks 05-01

  • Focus on graphical limit evaluation
  • Practice identifying discontinuity types
  • Master the limit evaluation strategies we covered today

See you next time for derivatives!

Appendix I: Practice Set B

Sketch of the piecewise function: x² + 1 for x < 0 ends at an open circle (0, 1), an isolated point sits at (0, 2), the line −x + 3 runs from an open circle (0, 3) down to (3, 0), and 6/(x − 3) falls away right of a dashed vertical asymptote at x = 3. Jump at x = 0, infinite discontinuity at x = 3.

Appendix II: Group Challenge

Three linear cost segments for small, medium and large batches. Total cost drops from 10,500 to 8,250 EUR at 500 units and from 30,750 to 25,500 EUR at 2,000 units, so both transitions are jump discontinuities.