What Is a Limit of a Function? Definition, Laws & Examples

A limit describes the value a function or sequence approaches as its input gets arbitrarily close to a specific point, without requiring the function to be defined at that point. Limits underpin calculus, providing the formal basis for continuity, derivatives, and integrals. When the limit exists, it is written as $$\lim_{x \to a} f(x) = L$$, meaning f(x) gets arbitrarily close to L as x approaches a.

Key Takeaways

• A limit exists only when a function approaches a single, well-defined value as the input nears a given point, whether that value is a real number or ±∞.
• One-sided limits, written $$\lim_{x \to a^+} f(x)$$ and $$\lim_{x \to a^-} f(x)$$, can differ from each other even when the two-sided limit does not exist.
• For the function f(x) = 1/x, the limit as x approaches 0 does not exist as a two-sided limit because the left-hand and right-hand limits go to negative and positive infinity respectively.
• A function is continuous at a point c only if both one-sided limits exist at c and equal f(c).
• Standard results such as $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ and $$\lim_{x \to 0} \frac{e^x – 1}{x} = 1$$ are used repeatedly in calculus computations.

What Is a Function?

A function defines a specific relationship between two sets: the domain (inputs) and the codomain (possible outputs). Given an element of the domain, a function assigns exactly one element of the codomain.

Key characteristics of a function:

1. Input-output relationship: Each input (independent variable) maps to a single output (dependent variable).
2. Notation: Functions are commonly written as f(x) or y = f(x).
3. Domain: The set of all input values for which the function is defined.
4. Codomain and range: The codomain is the set of possible outputs; the range is the set of actual outputs produced from the domain.
5. Uniqueness: A well-defined function never assigns multiple outputs to the same input.
6. Graph: Functions can be represented visually; the graph shows the mapping from input to output.
7. Examples: Linear, quadratic, trigonometric, and exponential functions.

How Is the Limit of a Function Defined?

A limit describes the value a function seems to approach when its argument approaches some point, even where the function is undefined at that point or grows without bound. When a function’s argument tends to a value c, this is written as:

$$\lim_{x \to c} f(x) = m$$

This is read as “as x tends to c, f(x) tends to m.” The statement does not assert the value of f(c); it only describes the values of f(x) when x is very close to c. If f is defined and continuous at c, then the limit equals the function’s value at that point:

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

What Are the Standard Limit Laws?

The following identities hold whenever the individual limits on the right-hand side exist:

$$\lim_{x \to c} \bigl(f(x) + g(x)\bigr) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$

$$\lim_{x \to c} \bigl(f(x)\cdot g(x)\bigr) = \bigl(\lim_{x \to c} f(x)\bigr)\cdot\bigl(\lim_{x \to c} g(x)\bigr)$$

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \quad \text{(when } \lim_{x\to c} g(x)\ne 0\text{)}$$

$$\lim_{x \to c} \bigl(\lambda\, f(x)\bigr) = \lambda\,\lim_{x \to c} f(x)$$

How Does Simplifying a Removable Discontinuity Work?

Consider the function $$f(x) = \frac{x^2 – 4}{x – 2}$$. Algebraically this simplifies to:

$$f(x) = \frac{(x – 2)(x + 2)}{x – 2} = x + 2, \quad x\ne 2$$

Because cancellation removes the factor x-2, f equals x + 2 for every x except x = 2. At x = 2 the original expression is undefined, since 0/0 is indeterminate, but values of f(x) near x = 2 approach 4. This gives:

$$\lim_{x \to 2} f(x) = 4$$

Since f(2) does not exist, $$\lim_{x \to 2} f(x) \ne f(2)$$, and f is not continuous at x = 2. If a new function g is defined by setting g(2) = 4, then g would be continuous at that point.

What Happens When a Limit Tends to Infinity?

The function $$f(x) = \frac{1}{x}$$ illustrates a limit that grows without bound. As x approaches 0, the absolute value of f(x) increases indefinitely, so the two-sided limit does not exist. The one-sided limits, however, are well-defined:

$$\lim_{x \to 0^+} \frac{1}{x} = +\infty \quad\text{and}\quad \lim_{x \to 0^-} \frac{1}{x} = -\infty$$

What Are Right-Hand and Left-Hand Limits?

A value c can be approached from the right, where x is greater than c, or from the left, where x is less than c. Writing x = c + h with h approaching 0 gives the right-hand approach; writing x = c − h with h approaching 0 gives the left-hand approach.

The right-hand limit (RHL) is written: $$\lim_{x \to c^+} f(x) = \lim_{h \to 0^+} f(c + h)$$

The left-hand limit (LHL) is written: $$\lim_{x \to c^-} f(x) = \lim_{h \to 0^+} f(c – h)$$

For a function to be continuous at c, both one-sided limits must exist and must equal f(c).

What Are Some Common Standard Limits?

Frequently used limit identities in calculus

Expression	Result
$$\lim_{x \to c} x$$	$$c$$
$$\lim_{x \to 0^+} \frac{1}{x}$$	$$+\infty$$ (two-sided limit does not exist)
$$\lim_{x \to \infty} x$$	$$\infty$$
$$\lim_{x \to 0} \frac{\sin x}{x}$$	$$1$$
$$\lim_{x \to 0} \frac{\tan x}{x}$$	$$1$$
$$\lim_{x \to 0} \frac{1 – \cos x}{x}$$	$$0$$
$$\lim_{x \to a} \frac{x^n – a^n}{x – a}$$	$$n a^{\,n-1}$$
$$\lim_{x \to a} \frac{\sin(x-a)}{x-a}$$	$$1$$
$$\lim_{x \to 0} \frac{\log_e(1 + x)}{x}$$	$$1$$
$$\lim_{x \to 0} \frac{a^x – 1}{x}$$	$$\log_e a$$
$$\lim_{x \to 0} \frac{e^x – 1}{x}$$	$$1$$

This discussion covers pointwise limits of functions. Limits of sequences of functions and other functional limits, as studied in functional analysis, are a separate topic not addressed here.

Frequently Asked Questions

Does a limit require the function to be defined at that point?

No. A limit describes the value a function approaches as its input nears a point, even if the function itself is undefined there. For example, $$\frac{x^2-4}{x-2}$$ is undefined at x = 2, yet its limit as x approaches 2 equals 4.

What does it mean for a two-sided limit not to exist?

A two-sided limit fails to exist when the left-hand and right-hand limits differ, or when one or both grow without bound in opposite directions. For $$f(x) = 1/x$$ at x = 0, the right-hand limit is +∞ and the left-hand limit is −∞, so no single two-sided limit exists.

How does a limit relate to continuity?

A function is continuous at a point c if the limit as x approaches c exists, both one-sided limits are equal, and that shared value equals f(c). If any of these conditions fail, the function is discontinuous at that point.

Can you cancel factors to simplify a limit calculation?

Yes, when a factor causing an indeterminate 0/0 form can be algebraically canceled, as with $$(x-2)$$ in $$\frac{x^2-4}{x-2}$$. After cancellation, the simplified expression, $$x+2$$ in this case, gives the correct limiting value even though the original expression was undefined at that point.

What is the difference between a limit and a functional limit in functional analysis?

The limits described here are pointwise limits of a single function’s output values as its input approaches a point. Functional analysis considers limits of sequences of functions themselves, a distinct and more advanced topic.