Limits

Introduction

1.Intuitively, the limit of a function $f(x)$ as $x$ approaches a point $x_0$ describes how the function behaves as the values of $x$ get arbitrarily close to $x_0$, without necessarily reaching that point. A neighbourhood of $x$ is an interval consisting of all points sufficiently close to $x$. More formally, a neighbourhood of $x$ is any open interval $(x - \delta, x + \delta)$ with $\delta > 0$. Limits are defined through neighbourhoods, and the local behaviour of a function near a point is described in the same terms.

2.The smaller the neighbourhood, the closer the points are to $x$. As the interval $(x - \delta, x + \delta)$ becomes narrower, that is, as $\delta$ approaches zero, the distance between the points of the neighbourhood and $x$ decreases accordingly.

3.The passage from average to instantaneous velocity is a physical example of this limiting process. Average velocity is defined over a nonzero time interval, while its limit as the interval shrinks, when that limit exists, is the velocity at a single instant.

Definition

4.Let $f(x)$ be a function whose behaviour we wish to study as $x$ approaches the point $x_0$. We say that, as $x$ tends to $x_0$, the function $f(x)$ has limit $\ell$, and we write:

$$\lim_{x \to x_0} f(x) = \ell$$

5.Formally, this statement asserts that for every tolerance $\varepsilon > 0$, there exists a corresponding distance $\delta > 0$ such that, whenever:

$$0 < |x - x_0| < \delta$$

6.it follows that:

$$|f(x) - \ell| < \varepsilon$$

7.Equivalently, for every neighbourhood of $\ell$ there is a neighbourhood of $x_0$ whose points, with the possible exception of $x_0$ itself, are all mapped into the neighbourhood of $\ell$.

8.When the definition is applied only to a right neighbourhood or only to a left neighbourhood of $x_0$, we refer to the right-hand limit and the left-hand limit, respectively. They are denoted as follows:

$$\lim_{x \to x_0^+} f(x) \quad \text{and} \quad \lim_{x \to x_0^-} f(x)$$

Asymptotes and infinite limits

9.In general, the value of $x$ in a limit can approach a real number $x_0$ or $\pm \infty$:

$$\lim_{x \to x_0} f(x) = \ell \quad \text{or} \quad \lim_{x \to x_0} f(x) = \pm \infty$$

10.Additionally, the value of the limit itself can be either a finite number or $\pm \infty$:

$$\lim_{x \to \pm \infty} f(x) = \ell \quad \text{or} \quad \lim_{x \to \pm \infty} f(x) = \pm \infty$$

11.When the limit of $f(x)$ exists and tends to $\pm \infty$ as $x$ approaches a finite real number $x_0$, the behaviour of the function near that point determines a vertical asymptote of equation $x = x_0$.

12.Often the two one-sided limits diverge in opposite directions:

$$\lim_{x \to x_0^+} f(x) = -\infty \quad \text{and} \quad \lim_{x \to x_0^-} f(x) = +\infty$$

13.When the limit of $f(x)$ exists and approaches a finite value $L$ as $x$ tends to $\pm\infty$, the line $y = L$ is a horizontal asymptote of the function.

14.This occurs when both limits at infinity equal the same value $L$:

$$\lim_{x \to +\infty} f(x) = L \quad \text{and} \quad \lim_{x \to -\infty} f(x) = L$$

15.

An asymptote is a line that the graph of a function approaches arbitrarily closely as either the $x$-value or the $y$-value increases or decreases without bound. The distance between the curve and the asymptote tends to zero as the graph extends toward the extremes of the coordinate plane. A systematic treatment of horizontal, vertical, and oblique asymptotes is developed in the dedicated page.

Conditions for limit existence and continuity

16.When the left-hand and right-hand limits of a function both exist and are finite, but have different values $\ell_1 \neq \ell_2$, we have:

$$ \begin{cases} \lim\limits_{x \to x_0^-} f(x) = \ell_1 \in \mathbb{R} \\[6pt] \lim\limits_{x \to x_0^+} f(x) = \ell_2 \in \mathbb{R} \end{cases} \implies \nexists \lim\limits_{x \to x_0} f(x) $$

17.The two-sided limit does not exist, because the function approaches two distinct values depending on the direction of approach. The two one-sided limits, however, are well defined and finite when considered separately.

18.According to the uniqueness theorem of limits, if the limit of a function $f(x)$ as $x$ approaches $x_0$ exists, whether finite or infinite, then such a limit is unique. The statement can be formalised as:

$$\lim_{x \to x_0} f(x) = \ell \in \overline{\mathbb{R}} \implies \ell \text{ is unique}$$

19.If two finite values $\ell_1 \neq \ell_2$ both satisfied the definition, choosing $\varepsilon < |\ell_1 - \ell_2|/2$ would force the values of $f$ near $x_0$ into two disjoint neighbourhoods at once, which cannot happen.

20.Limits also define what it means for a function to be continuous. A function $y = f(x)$ is continuous at a point $x_0$ if the limit of the function as $x$ approaches $x_0$ exists, is finite, and equals the value of the function at that point:

$$\lim_{x \to x_0} f(x) = f(x_0)$$

Properties

21.Limits respect the algebraic operations, so the limit of a sum, product, quotient, or constant multiple can be obtained from the limits of the parts. The algebra of limits treats each rule with proofs and worked examples.

22.The limit of the product of a constant and a function is equal to the product of the constant and the limit of the function, provided the limit exists.

$$\lim_{x \to x_0} \big( c f(x) \big) = c \lim_{x \to x_0} f(x) = c \cdot \ell$$

23.Multiplying a function by a constant does not affect the process of taking the limit, other than scaling the result by that constant.

24.The limit of the algebraic sum of two functions is equal to the sum of their individual limits, provided both limits exist.

$$\lim_{x \to x_0} \big( f(x) + g(x) \big) = \lim_{x \to x_0} f(x) + \lim_{x \to x_0} g(x) = \ell_1 + \ell_2$$

25.The limits of each function can therefore be evaluated separately and then added. This rule is particularly useful when working with polynomials, with trigonometric functions such as sine and cosine, and with other common elementary expressions.

26.The limit of the product of two functions is equal to the product of their individual limits, provided both limits exist.

$$\lim\limits_{x \to x_0} \big( f(x) g(x) \big) = \lim\limits_{x \to x_0} f(x) \cdot \lim\limits_{x \to x_0} g(x) = \ell_1 \cdot \ell_2$$

27.The limit of the quotient of two functions is equal to the quotient of their individual limits, provided both limits exist and the limit of the denominator is not zero.

$$\lim\limits_{x \to x_0} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim\limits_{x \to x_0} f(x)}{\lim\limits_{x \to x_0} g(x)} = \frac{\ell_1}{\ell_2}$$

When standard properties do not apply

28.The properties stated above are valid only when all relevant limits exist and are finite, and the denominator remains nonzero. In practical applications it is common to encounter expressions where direct substitution produces an undefined result, such as:

$$\frac{0}{0} \qquad \frac{\infty}{\infty} \qquad \infty - \infty$$

29.Such expressions are classified as indeterminate forms. Resolving them requires specialised techniques that go beyond standard algebraic manipulation, including factorisation, asymptotic comparison, L'Hôpital's rule, and the use of Taylor expansions combined with little-o notation. A familiar example is the limit:

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

30.Direct substitution of $x = 0$ yields $\frac{0}{0}$, which is undefined. The quotient property cannot be applied because the limit of the denominator is zero. The correct value is $1$, one of the remarkable limits.

Limits of elementary functions

31.The elementary functions have simple limits at infinity, and the logarithm at $0^+$. Most computations of harder limits reduce to these standard values.

32.For the constant function $f(x) = k$ with $k \in \mathbb{R}$, we have:

$$ \begin{align} \lim_{x \to -\infty} k &= k \\[6pt] \lim_{x \to +\infty} k &= k \end{align} $$

33.For the identity function $f(x) = x$, we have:

$$ \begin{align} \lim_{x \to -\infty} x &= -\infty \\[6pt] \lim_{x \to +\infty} x &= +\infty \end{align} $$

34.For the exponential function with base $a > 1$, we have:

$$ \begin{align} \lim_{x \to -\infty} a^x &= 0 \\[6pt] \lim_{x \to +\infty} a^x &= +\infty \end{align} $$

35.For the exponential function with base $0 < a < 1$, we have:

$$ \begin{align} \lim_{x \to -\infty} a^x &= +\infty \\[6pt] \lim_{x \to +\infty} a^x &= 0 \end{align} $$

36.For the power function $f(x) = x^n$ with even exponent $n \in \mathbb{N}$, we have:

$$ \begin{align} \lim_{x \to -\infty} x^n &= +\infty \\[6pt] \lim_{x \to +\infty} x^n &= +\infty \end{align} $$

37.For the power function with odd exponent, we have:

$$ \begin{align} \lim_{x \to -\infty} x^n &= -\infty \\[6pt] \lim_{x \to +\infty} x^n &= +\infty \end{align} $$

38.For the root function $f(x) = \sqrt[n]{x}$ with even index, we have:

$$\lim_{x \to +\infty} \sqrt[n]{x} = +\infty$$

39.

For even indices, the root function is defined only for $x \geq 0$. The limit as $x \to -\infty$ is therefore not applicable.

40.For the root function with odd index, we have:

$$ \begin{align} \lim_{x \to -\infty} \sqrt[n]{x} &= -\infty \\[6pt] \lim_{x \to +\infty} \sqrt[n]{x} &= +\infty \end{align} $$

41.For the logarithmic function with base $a > 1$, we have:

$$ \begin{align} \lim_{x \to 0^+} \log_a x &= -\infty \\[6pt] \lim_{x \to +\infty} \log_a x &= +\infty \end{align} $$

42.For the logarithmic function with base $0 < a < 1$, we have:

$$ \begin{align} \lim_{x \to 0^+} \log_a x &= +\infty \\[6pt] \lim_{x \to +\infty} \log_a x &= -\infty \end{align} $$

43.For the absolute value function $f(x) = |x|$, we have:

$$ \begin{align} \lim_{x \to -\infty} |x| &= +\infty \\[6pt] \lim_{x \to +\infty} |x| &= +\infty \end{align} $$

44.For the sign function $\mathrm{sgn}(x)$, we have:

$$ \begin{align} \lim_{x \to -\infty} \mathrm{sgn}(x) &= -1 \\[6pt] \lim_{x \to +\infty} \mathrm{sgn}(x) &= 1 \end{align} $$