Limit of a Function

The limit (l) of a function f(x) as x approaches x₀
$$ \lim_{x \rightarrow x_0} f(x) = l $$ can be classified as:
• convergent if l is a finite number
• divergent if l is infinite
• nonexistent if the function oscillates or the limit fails to exist at x₀

Here, x₀ is an accumulation point of the domain of f(x), while l may be any real number, either finite or infinite.

Convergent Limit

A] When x₀ is a finite number

The limit of a function f(x) as x approaches x₀ in ℝ is equal to l
$$ \lim_{x \rightarrow x_0} f(x) = l $$ if and only if, for every ε > 0, there exists a δ > 0 such that:
$$ l - \epsilon < f(x) < l + \epsilon $$ for all x within the interval (x₀ - δ, x₀ + δ).

In this case, l is a finite real number.

In symbolic terms:

$$ \lim_{x \rightarrow x_0} f(x) = l \Leftrightarrow \forall \epsilon>0, \exists \delta>0: |f(x)-l|<\epsilon, \forall x \in A: 0 \ne |x-x_0|<\delta $$

Note. The limit of a function can also be characterized using subsequences. The limit of f(x) as x → x₀ converges to the finite number l within a neighborhood A - {x₀} if, for every subsequence x_n contained in A and converging to x₀, we have f(x_n) = l.

B] When x₀ approaches positive or negative infinity

The limit of a function f(x) as x tends to infinity is equal to the finite limit l
$$ \lim_{x \rightarrow \infty} f(x) = l $$ if and only if, for every ε > 0, there exists a number k > 0 such that:
$$ |f(x) - l| < \epsilon $$ for all x > k.

Here, l is a finite real number.

Symbolically:

$$ \lim_{x \rightarrow \infty} f(x) = l \Leftrightarrow \forall \epsilon>0, \exists k>0: |f(x)-l|<\epsilon, \forall x>k $$

The definition is analogous for x approaching -∞.

Example. Consider the function f(x) = 1/x. We compute its limit as x → ∞:
$$ \lim_{x \rightarrow +\infty} \frac{1}{x} = 0 $$

Divergent Limit to Infinity

A] When x₀ is a finite number

The limit of a function as x → x₀ diverges to infinity
$$ \lim_{x \rightarrow x_0} f(x) = \infty $$ if, for every real number M > 0, there exists a δ > 0 such that f(x) > M for all x within the interval (x₀ - δ, x₀ + δ).

The limit can diverge to positive infinity (+∞) or negative infinity (-∞).

Symbolically:

$$ \lim_{x \rightarrow x_0} f(x) = \infty \Leftrightarrow \forall M>0, \exists \delta>0: f(x)>M, \forall x \in A: 0 \ne |x - x_0| < \delta $$

Example. Consider the function f(x) = 1/(x - 1). We compute the limit as x → 1:
$$ \lim_{x \rightarrow 1} \frac{1}{x-1} = +\infty $$

B] When x₀ approaches positive or negative infinity

The limit of a function as x → +∞ diverges to infinity
$$ \lim_{x \rightarrow +\infty} f(x) = \infty $$ if, for every real number M > 0, there exists a number k > 0 such that f(x) > M for all x > k.

The limit can diverge to either +∞ or -∞.

Symbolically:

$$ \lim_{x \rightarrow +\infty} f(x) = \infty \Leftrightarrow \forall M>0, \exists k>0: f(x)>M, \forall x>k $$

The same definition applies when x → -∞.

Example. Consider the function f(x) = x². We compute the limit as x → ∞:
$$ \lim_{x \rightarrow +\infty} x^2 = \infty $$

Nonexistent Limit

A] When x₀ is a finite number

The limit of a function as x approaches x₀ does not exist if the right-hand and left-hand limits are not equal:
$$ \lim_{ x \rightarrow x_0^+ } f(x) \ne \lim_{ x \rightarrow x_0^- } f(x) $$

In such situations, the function might be undefined at x₀ or possess a vertical asymptote there.

B] When x₀ approaches positive or negative infinity

The limit of a function as x → ±∞ does not exist if the function oscillates indefinitely or is undefined:
$$ \lim_{ x \rightarrow \pm \infty } f(x) = \text{does not exist} $$

Examples

Consider the function:

$$ f(x) = \frac{1}{x} $$

The limit of this function as x → 0 does not exist:

$$ \lim_{ x \rightarrow 0 } \frac{1}{x} = \text{does not exist} $$

because the right-hand limit is +∞:

$$ \lim_{ x \rightarrow 0^+ } \frac{1}{x} = +\infty $$

while the left-hand limit is -∞:

$$ \lim_{ x \rightarrow 0^- } \frac{1}{x} = -\infty $$

The function 1/x is undefined at x₀ = 0, where it has a vertical asymptote.

And so forth.