# Continuity, Limits & L’Hôpital’s Rule (AA HL) | Maths with Lemon

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### Continuity, Differentiability, Limits & L’Hôpital’s Rule

IB Mathematics: Analysis and Approaches Higher Level · Lesson 10.6 · AHL 5.12

Understand the behaviour of functions near a point, differentiate from first principles, and evaluate indeterminate limits using L’Hôpital’s rule.

Lesson 10.6 worksheet (PDF) · Previous lesson: Maclaurin Series


#### 1. Continuity and Differentiability


##### What you need to know

- A function is continuous at a point when its limit equals its value there.
- A differentiable function has a well-defined derivative at the point.
- Differentiability implies continuity, but continuity does not imply differentiability.
- At a corner, the one-sided gradients differ.

##### Definitions

For \(f(x)=x^2\) when \(x\leq3\), and \(f(x)=x+a\) when \(x>3\), match the two expressions at \(x=3\).


#### 2. Differentiability of Piecewise Functions


##### Two conditions

- First match the function values from both sides.
- Then match the left and right derivatives.
- Both conditions are necessary for differentiability.

##### Worksheet example

Suppose \(f(x)=1/x\) for \(x\leq1\) and \(f(x)=ax+b\) for \(x>1\).


#### 3. Understanding Limits


##### Convergence and divergence

- A convergent limit approaches a finite value.
- A divergent expression does not approach a finite limit.
- For rational expressions at infinity, compare leading powers or divide by the dominant term.
- Always test direct substitution before choosing a technique.

##### Worked examples


#### 4. L’Hôpital’s Rule


##### When it applies

- The quotient must have an indeterminate form \(0/0\) or \(\infty/\infty\).
- Differentiate the numerator and denominator separately.
- Re-evaluate the limit after differentiating.
- Do not use the quotient rule to differentiate the whole fraction.
- The relevant differentiability and limit conditions must hold.

##### The rule

Use this equality when the original ratio is of an allowed indeterminate form and the derivative-ratio limit exists (possibly as an infinite limit), under the rule’s usual conditions.


#### 5. The Indeterminate Form 0/0


##### Method checklist

- Substitute to confirm \(0/0\).
- Differentiate numerator and denominator.
- Substitute again.
- If still indeterminate, consider another application.

##### Worksheet examples


#### 6. The Indeterminate Form Infinity/Infinity


##### Limits as \(x\to\infty\)

- Identify the growth rates of the numerator and denominator.
- Confirm an infinity-over-infinity form.
- Apply L’Hôpital’s rule if appropriate.
- Compare with an algebraic simplification where possible.

##### Worked examples


#### 7. Rearranging Other Indeterminate Forms


##### Before using the rule

- A product such as \(0\cdot\infty\) is not directly eligible.
- Rewrite a product as a quotient.
- A difference such as \(\infty-\infty\) may require a common denominator or rationalisation.
- Pay attention to one-sided limits and domains.

##### Worksheet example


#### 8. Repeated Use of L’Hôpital’s Rule


##### Key idea

- After one differentiation, check the new form.
- If it is still \(0/0\) or \(\infty/\infty\), the rule may be applied again.
- Stop as soon as the limit can be evaluated.
- Maclaurin series provide an alternative approach.

##### Worked example


#### 9. Interactive Limit Explorer


##### Investigate the behaviour near a point

- Choose one of three limits from the lesson.
- Adjust the distance \(h\) from the target.
- Compare the function value with the predicted limit.
- Observe how numerical approximations become unstable at extremely small distances.

##### Numerical approach to a limit


#### 10. HL Practice and Common Mistakes


##### Practice from Lesson 10.6

- Find limits involving \(e^x\), \(\ln x\), \(\sin x\), and radicals.
- Determine whether expressions converge or diverge.
- Investigate \(x e^x\) as \(x\to-\infty\).
- Evaluate \(1/\sin x-1/x\) as \(x\to0\).
- Prove the worksheet’s final challenge involving \(e^x\) and \(e^{-x}\).

##### Common mistakes

- Using L’Hôpital’s rule before checking the indeterminate form.
- Applying the quotient rule rather than differentiating top and bottom separately.
- Assuming all infinite limits have finite values.
- Ignoring whether a limit is one-sided.
- Forgetting to test whether repeated differentiation is necessary.
Open the complete worksheet


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