2.1 Limit and its Relation to Differentiation
The concept of limits is fundamental to calculus and differential operations. Just as the speedometer reading of a moving car gives its instantaneous velocity at a specific moment through approximation, limits allow us to examine the behavior of a function \(f(x)\) as its variable \(x\) approaches a specific value \(a\). Furthermore, the idea of limits serves as the foundation for defining the gradient of a tangent to a curve, leading directly to the definition of the first derivative of a function by first principles.
Learning Objectives
- Determine the limit of a function \(f(x)\) as its variable approaches a real value \(a\), written as \(\lim_{x \to a} f(x) = L\).
- Evaluate limits involving indeterminate forms \(\left(\frac{0}{0}\right)\) by using factorisation or rationalisation of the numerator or denominator.
- Understand the relationship between the gradient of a secant line (chord) and the gradient of a tangent line as \(\delta x \to 0\).
- Determine the first derivative \(\frac{dy}{dx}\) of a function \(y = f(x)\) using first principles: \(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}\).
Flowchart for Finding the Limit of a Function \(\lim_{x \to a} f(x)\) and the Geometry of Secant to Tangent
A clear decision pathway when evaluating limits and a geometric view of how the chord \(BC\) turns into the tangent line as \(\delta x \to 0\).
Limit Evaluation Flowchart
Substitute \(x = a\) into \(f(x)\).
If the result is not \(\frac{0}{0}\), the limit is \(f(a)\).
If the result is \(\frac{0}{0}\), use factorisation or conjugate rationalisation before substituting again.
Secant to Tangent Geometry
Points \(B(x, y)\) and \(C(x + \delta x, y + \delta y)\) lie on the curve \(y = f(x)\).
The gradient of secant \(BC\) is \(\frac{\delta y}{\delta x}\).
As \(\delta x \to 0\), the secant line approaches the tangent line at \(B\).
Short Explanation
A limit answers the question: what value does the output of a function approach as the input gets infinitely close to a certain target value?
Concept and Notation of Limits
When a variable \(x\) approaches a value \(a\) (where \(x \neq a\)), the value of \(f(x)\) approaches a value \(L\). This is written mathematically as:
\[\lim_{x \to a} f(x) = L\]
A limit exists if the function values approach the same number \(L\) as \(x\) approaches \(a\) from both the left (\(x < a\)) and the right (\(x > a\)).
Undefined vs Limit Value
A function \(f(x)\) may not be defined at \(x = a\), but its limit as \(x \to a\) can still exist.
Example: \(f(x) = \frac{x^2 + 3x}{x}\) at \(x = 0\) gives \(\frac{0}{0}\), but \(\lim_{x \to 0} \frac{x^2 + 3x}{x} = 3\).
| Step | Action | Result |
|---|---|---|
| 1 | Substitute \(x = a\) directly into \(f(x)\). | \(f(a)\) or \(\frac{0}{0}\) |
| 2 | If \(f(a) \neq \frac{0}{0}\), the limit is \(f(a)\). | Direct substitution works. |
| 3 | If \(f(a) = \frac{0}{0}\), use factorisation or rationalisation. | Cancel the zero-producing term, then substitute. |
Four-Step Process to Differentiate by First Principles
Consider two points \(A(x, y)\) and \(B(x + \delta x, y + \delta y)\) on the curve \(y = f(x)\).
Find \(\delta y = f(x + \delta x) - f(x)\) and expand/simplify the expression.
Form the ratio \(\frac{\delta y}{\delta x}\) by dividing both sides by \(\delta x\).
Take the limit as \(\delta x \to 0\): \(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x}\).
First derivative by first principles
\(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}\)\(\frac{dy}{dx}\) is a single calculus symbol representing this limit, not a fraction dividing \(dy\) by \(dx\).
Gradient of tangent at \(B\)
\(\text{Gradient of tangent at } B = \lim_{\delta x \to 0} \frac{\delta y}{\delta x}\)The secant line \(BC\) approaches the tangent line at \(B\) as \(\delta x \to 0\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Evaluate \(\lim_{x \to 1} \frac{x^2 - 1}{x - 1}\).
Evaluate \(\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}\).
Find \(\frac{dy}{dx}\) for the function \(y = 3x^2\) by using first principles.
Common Mistakes
- Treating \(\frac{0}{0}\) as equal to 0, 1, or undefined without further analysis. The form \(\frac{0}{0}\) is indeterminate, meaning the limit must be evaluated by factorisation or rationalisation.
- Dropping the limit notation \(\lim\) prematurely before the substitution step has actually been performed.
- Misinterpreting the notation \(\frac{dy}{dx}\) as \(dy\) divided by \(dx\) that can be split algebraically at this introductory level, rather than recognising it as the notation for the limit \(\lim_{\delta x \to 0} \frac{\delta y}{\delta x}\).
- Algebraic errors in expanding \((x + \delta x)^2\) or \((x + \delta x)^3\), such as writing \((x + \delta x)^2\) as \(x^2 + (\delta x)^2\), leaving out the middle cross-term \(2x(\delta x)\).
- Sign errors when multiplying by conjugates, particularly forgetting to distribute negative signs inside brackets when applying \((a - b)(a + b) = a^2 - b^2\).
Concept Misunderstandings
If \(f(a)\) does not exist, then \(\lim_{x \to a} f(x)\) cannot exist.
The value of \(f(a)\) describes the function at the exact point \(x = a\), whereas \(\lim_{x \to a} f(x)\) describes what value the function approaches as \(x\) gets arbitrarily close to \(a\) without necessarily reaching \(a\). A limit can exist even when the point itself has a "hole" or is undefined.
\(\delta x\) is equal to 0.
\(\delta x\) approaches 0 (\(\delta x \to 0\)), meaning it gets infinitely small, but it is not 0 during the division stage \(\frac{\delta y}{\delta x}\), which is why dividing by \(\delta x\) is mathematically valid.
The gradient of a tangent is the same as the gradient between any two distant points on a curve.
The gradient of a secant line between two distant points is an average rate of change. The gradient of the tangent is the instantaneous rate of change at that single point, obtained only when \(\delta x \to 0\).
Summary
- The notation \(\lim_{x \to a} f(x) = L\) indicates that as \(x\) approaches \(a\), \(f(x)\) approaches \(L\).
- If direct substitution produces an indeterminate form \(\frac{0}{0}\), evaluate the limit using factorisation or rationalisation.
- The gradient of a secant line between \(B(x, y)\) and \(C(x + \delta x, y + \delta y)\) is \(\frac{\delta y}{\delta x}\).
- As \(\delta x \to 0\), the secant line becomes the tangent line at \(B\).
- The first derivative by first principles is defined as: \(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}\).
Short Activity
Objective: Identify limits, indeterminate forms, first principles and key terms in differentiation.
A. Objective Quiz
1 What is the value of \(\lim_{x \to 4} \frac{3 - \sqrt{x}}{x + 2}\)?
2 In differentiation from first principles, what does \(\lim_{\delta x \to 0} \frac{\delta y}{\delta x}\) represent geometrically?
B. Fill in the Blanks
3 If direct substitution of \(x = a\) into a rational function yields \(\frac{0}{0}\), this result is known as an __________ form.
4 The definition of the first derivative using first principles is written as \(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\text{________}}\).
C. Matching / Drag and Drop
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
