Intuitive Idea of Limits

Before formalizing the concept, it is important to understand the intuitive idea of a limit.

A limit describes the behavior of a function when the variable approaches a value, without necessarily reaching it.

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Intuition

When we write:

$$\lim_{x \to a} f(x)$$

we want to know:

what value $f(x)$ approaches when $x$ approaches $a$

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Classic Example

Consider the function:

$$f(x) = \frac{x^2 - 1}{x - 1}$$

If we substitute $x = 1$, we get:

$$\frac{0}{0}$$

an indeterminate expression

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But We Can Simplify

$$\frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x-1}$$ $$= x + 1 \quad (x \neq 1)$$

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Now Calculate the Limit

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} (x + 1) = 2$$

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Interpretation

Even though the function is not defined at $x = 1$, the values of $f(x)$ approach:

$$2$$

This is the limit

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Visually

• $x$ approaches 1
• $f(x)$ approaches 2

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Important Example

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

This is one of the most important limits in calculus.

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Be Careful

A limit is not simply substituting the value.

Sometimes:

• it works (continuous functions)
• sometimes it produces an indeterminate form: $\frac{0}{0}$

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Summary

• A limit describes behavior, not an exact value
• We can approach a point without reaching it
• Indeterminate forms require algebraic manipulation