Limits and continuity

Limits

A limit describes where outputs go as inputs approach a point. Compare the two sides before deciding whether a single limit exists.

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1. Approach from both sides

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Example graph

f(x)=x+1,\quad \lim_{x\to1}f(x)=2

For f(x)=x+1, sample x=1−h and x=1+h. As the positive distance h shrinks, both heights approach 2. The two-sided limit is 2.

f(1−h): 1
f(1+h): 3

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Marked points show sampled inputs. Open circles exclude a point. The graph is a finite window; a limit is a statement about approaching, not a single sample.

2. A limit through a hole

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Example graph

f(x)=\frac{x^2-1}{x-1},\quad x\ne1

Cancel x−1 only for x≠1. The original expression has no value at 1, but its nearby values follow x+1 and approach 2.

f(1−h): 1
f(1+h): 3

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Marked points show sampled inputs. Open circles exclude a point. The graph is a finite window; a limit is a statement about approaching, not a single sample.

3. When the sides disagree

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Example graph

f(x)=\begin{cases}1&x<0\\2&x\ge0\end{cases}

Here f(x)=1 for x<0 and f(x)=2 for x≥0. Shrinking the distance cannot make the heights agree. There is no two-sided limit at zero.

f(−h): 1
f(h): 2

Marked points show sampled inputs. Open circles exclude a point. The graph is a finite window; a limit is a statement about approaching, not a single sample.

4. As the input grows

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Example graph

f(x)=\frac{x}{x+1},\quad\lim_{x\to\infty}f(x)=1

For positive x, x/(x+1)=1−1/(x+1). The difference from 1 shrinks as x grows without bound. The graph approaches 1 without needing to reach it.

f(a): 0.5
1−f(a): 0.5

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Marked points show sampled inputs. Open circles exclude a point. The graph is a finite window; a limit is a statement about approaching, not a single sample.