Limits and continuity

Asymptotes

Study lines that describe a curve near an excluded input or far away. An asymptote is not an impenetrable wall.

1. A vertical asymptote

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Example graph

f(x)=\frac1{x-1}

For 1/(x−1), approaching 1 from the right sends values upward without bound; from the left they decrease without bound. No finite value at 1 repairs this.

f(1−h): -1
f(1+h): 1

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 horizontal asymptote

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Example graph

f(x)=2+\frac1x,\quad y=2

The term 1/x tends to zero for large positive or negative x, so 2+1/x approaches y=2. Other functions can cross their horizontal asymptotes.

f(a): 3
|f(a)−2|: 1

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. An oblique asymptote

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Example graph

f(x)=x+\frac1x,\quad f(x)-x=\frac1x

For x+1/x, the difference from the line y=x is 1/x. That difference tends to zero even though both the curve and line are unbounded.

f(a): 2
f(a)−a: 1

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.