Maths with Lemon

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

\[\lim_{x\to a}f(x)=f(a)\]
\[f^{\prime}(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]
Worksheet: continuity at \(x=3\)

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

\[9=3+a\quad\Longrightarrow\quad\boxed{a=6}\]

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

Find \(a\) and \(b\) at \(x=1\)

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

\[a+b=1\qquad\text{(continuity)}\]
\[a=-1\qquad\text{(matching gradients)}\]
\[\boxed{a=-1,\quad b=2}\]

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

Exponential ratio
\[\lim_{x\to\infty}\frac{e^x-1}{2e^x+1}\]
\[=\lim_{x\to\infty}\frac{1-e^{-x}}{2+e^{-x}}=\boxed{\frac12}\]
Divergence
\[\lim_{x\to\infty}\frac{x^2}{1+3x}=+\infty\]

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

\[\boxed{\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)}}\]

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.

Classic trigonometric limit
\[\lim_{x\to0}\frac{\sin x}{x}=\lim_{x\to0}\frac{\cos x}{1}=\boxed{1}\]

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

Exponential and sine
\[\lim_{x\to0}\frac{e^x-1}{\sin x}=\lim_{x\to0}\frac{e^x}{\cos x}=\boxed{1}\]
Square-root expression
\[\lim_{x\to1}\frac{\sqrt{x}-1}{x-1}\]
\[=\lim_{x\to1}\frac{1/(2\sqrt{x})}{1}=\boxed{\frac12}\]

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

Linear over linear
\[\lim_{x\to\infty}\frac{10x+3}{5x+1}=\lim_{x\to\infty}\frac{10}{5}=\boxed{2}\]
Logarithm over a linear function
\[\lim_{x\to\infty}\frac{\ln x}{x}=\lim_{x\to\infty}\frac{1/x}{1}=\boxed{0}\]

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

Evaluate \(\lim_{x\to0^+}x\ln x\)
\[x\ln x=\frac{\ln x}{1/x}\]
\[\lim_{x\to0^+}\frac{\ln x}{1/x}=\lim_{x\to0^+}\frac{1/x}{-1/x^2}\]
\[=\lim_{x\to0^+}(-x)=\boxed{0}\]

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

A limit requiring two applications
\[\lim_{x\to0}\frac{\cos x-1}{x^2}\]
\[=\lim_{x\to0}\frac{-\sin x}{2x}\]
\[=\lim_{x\to0}\frac{-\cos x}{2}=\boxed{-\frac12}\]

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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