55  Limits and Continuity

55.1 Introduction

Limits and Continuity are the cornerstones of calculus.
- A limit describes the value a function approaches as the input approaches a particular point.
- Continuity ensures the function has no breaks, jumps, or holes.

Mastering limits and continuity helps in solving calculus problems, evaluating indeterminate forms, and preparing for differentiation/integration.

55.2 Limits

55.2.1 Definition

If \(f(x)\) approaches a finite value \(L\) as \(x\) approaches \(a\), then \[ \lim_{x\to a} f(x)=L \]

55.2.2 Left-hand and Right-hand Limits

  • Left-hand limit: \(\lim_{x\to a^-} f(x)\) (approaching from left).
  • Right-hand limit: \(\lim_{x\to a^+} f(x)\) (approaching from right).
    If both exist and are equal, then \(\lim_{x\to a} f(x)\) exists.

55.2.3 Standard Limits

  1. \(\lim_{x\to 0}\frac{\sin x}{x}=1\)
  2. \(\lim_{x\to 0}\frac{\tan x}{x}=1\)
  3. \(\lim_{x\to 0}\frac{1-\cos x}{x^2}=\tfrac{1}{2}\)
  4. \(\lim_{x\to 0}(1+x)^{1/x}=e\)
  5. \(\lim_{x\to 0}\frac{\ln(1+x)}{x}=1\)
  6. \(\lim_{x\to\infty}\Big(1+\frac{1}{x}\Big)^x=e\)

55.2.4 Indeterminate Forms

  • \(\frac{0}{0}, \frac{\infty}{\infty}, 0\cdot\infty, \infty-\infty, 0^0, 1^\infty, \infty^0\).
  • These require algebraic simplification, L’Hôpital’s Rule, or series expansion.

55.2.5 L’Hôpital’s Rule

If \(\lim_{x\to a} f(x)/g(x)\) gives \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), then \[ \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} \] (provided the latter limit exists).

55.3 Continuity

55.3.1 Definition

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 these conditions hold for every \(a\) in domain, \(f(x)\) is continuous on that interval.

55.3.2 Types of Discontinuity

  1. Removable: hole in the curve (limit exists but \(f(a)\) missing/mismatched).
  2. Jump: left-hand and right-hand limits exist but are unequal.
  3. Infinite: function tends to \(\pm\infty\) at some point.

55.3.3 Important Theorems

  1. If \(f\) and \(g\) are continuous at \(a\), then \(f\pm g, f\cdot g, f/g\) (if \(g(a)\neq0\)) are continuous.
  2. Composition of continuous functions is continuous.
  3. Polynomial and exponential functions are continuous everywhere.
  4. Trigonometric functions are continuous in their domains.

55.4 Applications

  • Evaluating tricky limits.
  • Checking continuity before applying differentiation.
  • Determining asymptotic behavior.
  • Understanding function behavior in economics, physics, and probability.

55.5 Solved Examples

Example 1
Evaluate \(\lim_{x\to 0}\frac{\sin 5x}{x}\).
= \(5\lim_{x\to 0}\frac{\sin 5x}{5x}=5(1)=5\).

Example 2
Evaluate \(\lim_{x\to 0}\frac{e^x-1}{x}\).
= \(1\).

Example 3
Evaluate \(\lim_{x\to 0}\frac{\tan x-\sin x}{x^3}\).
Expand using series: \(\tan x=x+\tfrac{x^3}{3}+\cdots\), \(\sin x=x-\tfrac{x^3}{6}+\cdots\).
Numerator ≈ \(x+\tfrac{x^3}{3}-x+\tfrac{x^3}{6}=\tfrac{x^3}{2}\).
So limit = \(\tfrac{1}{2}\).

Example 4
Check continuity of \(f(x)=\begin{cases} x^2, & x\leq 1 \\ 2x-1, & x>1 \end{cases}\) at \(x=1\).
- Left limit at 1 = \(1^2=1\).
- Right limit at 1 = \(2(1)-1=1\).
- \(f(1)=1\).
Thus continuous at \(x=1\).

Example 5
Check continuity of \(f(x)=\frac{1}{x}\) at \(x=0\).
Function undefined at \(x=0\) → discontinuous.

Example 6
Evaluate \(\lim_{x\to \infty}\Big(1+\frac{2}{x}\Big)^x\).
= \(e^2\).

55.6 Practice Problems

  1. Evaluate \(\lim_{x\to 0}\frac{\sin 7x}{x}\).
  2. Evaluate \(\lim_{x\to 0}\frac{1-\cos x}{x^2}\).
  3. Check continuity of \(f(x)=|x|\) at \(x=0\).
  4. Evaluate \(\lim_{x\to 0}\frac{\ln(1+2x)}{x}\).
  5. Evaluate \(\lim_{x\to 0}\frac{x}{\tan x}\).
  6. Check continuity of \(f(x)=\frac{x^2-1}{x-1}\) at \(x=1\).
  7. Evaluate \(\lim_{x\to \infty}\frac{2x^2+3}{x^2-1}\).
  8. Determine type of discontinuity at \(x=0\) for \(f(x)=\frac{\sin x}{x}\).
  9. Check continuity of \(f(x)=\sin(1/x)\) at \(x=0\).
  10. Evaluate \(\lim_{x\to 0}\frac{e^{2x}-1}{x}\).

55.7 Answer Key (Concise)

  1. \(7\)
  2. \(1/2\)
  3. Continuous (both sides give 0).
  4. \(2\)
  5. \(1\)
  6. Simplify \(\frac{x^2-1}{x-1}=x+1\) (for \(x\neq1\)). Limit at 1 = 2. But \(f(1)\) undefined → removable discontinuity.
  7. \(\lim=2\)
  8. Continuous (limit exists and =1).
  9. Not continuous (oscillates).
  10. \(2\)

55.8 Summary

  • Limit describes approach; continuity ensures smoothness.
  • Standard limits and L’Hôpital’s rule essential for exams.
  • Types of discontinuity: removable, jump, infinite.
  • Continuous functions behave predictably and support differentiation.