CHAPTER 03

Change and Limits

Limits are one of the central ideas of calculus. This chapter develops the idea carefully through graphical, numerical and algebraic approaches, then studies important types of limiting behaviour.

\[x\to a\quad\Longrightarrow\quad f(x)\to L\quad\Longrightarrow\quad\lim_{x\to a}f(x)=L\]

CHANGE AND LIMITS

Understanding limits by type

The Idea of a Limit

Understand what approaching a value means using numerical examples, function graphs and algebraic expressions.

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One-Sided Limits

Approach a point from the left and from the right, and determine when a two-sided limit exists.

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Limit Laws and Algebraic Methods

Evaluate limits using substitution, limit laws, factorisation, cancellation and rationalisation.

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Infinite Limits

Study functions that become unbounded near a finite input and understand vertical asymptotes.

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Limits at Infinity

Study long-run behaviour as \(x\to\infty\) or \(x\to-\infty\), including horizontal asymptotes.

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Why limits matter

\[\frac{f(x+h)-f(x)}{h}\quad\xrightarrow{\ h\to0\ }\quad f'(x)\]

Limits provide the mathematical bridge from finite change to instantaneous rate of change. Continuity is developed separately in the next chapter before differentiation.