Limit of a Function

What Is a Limit of a Function? A 5 Minute Introduction

Definition/Summary

Limits are a mathematical tool that is used to define the ‘limiting value’ of a function i.e. the value a function seems to approach when its argument(s) approach a particular value. Although the argument of the function can be taken to approach any value, limits are helpful in cases where the argument approaches a value where the function is not defined or becomes exceedingly large.

While defining a limit, we say that the argument ‘tends to’ a value. For example,

[tex]\lim_{x \to c}f(x) = m[/tex]

is said: “As x tends to c, the function f(x) tends to m”. This statement however makes no assertion of what the value of f(c) would be. Rather, it means that as x becomes exceedingly close to ‘c’, f(x) becomes exceedingly close to m.

If, however, the function is defined and continuous at ‘c’, then:

[tex]\lim_{x \to c}f(x) = f(c)[/tex]

(See explanation)

Equations

Identities of limits:

[tex]\lim_{x \to c}f(x) + g(x) = \lim_{x \to c}f(x) + \lim_{x \to c}g(x)[/tex]

[tex]\lim_{x \to c}f(x) \cdot g(x) = \lim_{x \to c}f(x) \cdot \lim_{x \to c}g(x)[/tex]

[tex]\lim_{x \to c}\frac{f(x)}{g(x)} = \frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)}[/tex]

[tex]\lim_{x \to c}\lambda f(x) = \lambda\,\lim_{x \to c} f(x)[/tex]

Extended explanation

1. Informal approach to understanding limits

Let there be a function f(x) such that:

[tex]f(x) = \frac{x^2 – 4}{x – 2}[/tex]

This function can be simplified to:

[tex]f(x) = \frac{(x – 2)(x + 2)}{x – 2} = x + 2[/tex]

However, when (x – 2) is cancelled in both the numerator and the denominator, x – 2 ≠ 0 or x ≠ 2. This is because [itex]\frac{0}{0}[/itex] is an indeterminate form. Hence, the function f(x) = x + 2 at points other than x = 2.

But, the function is still defined at points very close to 2. As x gets closer and closer to 2, f(x) gets closer and closer to 4. So, it is said that f(x) tends to 4 as x tends to 2. This is written as:

[tex]\lim_{x \to 2} f(x) = 4[/tex]

This function, however is not defined at x = 2 i.e. f(2) does not exist at all and hence,

[tex]\lim_{x \to 2} f(x) \ne f(2)[/tex]

and hence, the function is said to be non-continuous at x = 2. This can be understood as there being a gap on the line plot of this function.

However, consider the function g(x) = x + 2. It is defined at all points, and hence is continuous everywhere. Other than for their behavior at x = 2, f(x) and g(x) are identical functions.

Even g(x) can be made to be continuous by describing it as:

[tex]g(x) = \begin{cases}x + 2, & x \ne 2 \\0, & x = 2\end{cases}[/tex]

Here, g(2) = 0, and hence:

[tex]\lim_{x \to 2} g(x) \ne g(2)[/tex]

and hence the function is not continuous at x = 2. The following function however, is continuous at x = 2:

[tex]g(x) = \begin{cases}x + 2, & x \ne 2 \\4, & x = 2\end{cases}[/tex]

For more info, see the article on Continuity.

2. Tending to infinity

Consider the function f(x) such that:

[tex]f(x) = \frac{1}{x}[/tex]

As ‘x’ gets close to zero, the value of f(x) becomes large. At x = 0, f(x) is not defined. However, f(x) ‘tends to’ infinity, as ‘x’ tends to zero:

[tex]\lim_{x \to 0} f(x) = \infty[/tex]

3. Right hand and Left hand limits

In a graph of a function, when we approach a particular value, we can do it either from the right side or from the left side of the graph. Mathematically, when we say that, ‘x tends to c’, it can mean either that x is infinitesimally larger than c or it is infinitesimally smaller than c’.

[tex]\lim{x \to c}f(x)[/tex]

Here, let:

[tex]x = c + h[/tex]

and

[tex]as~~x \to c~~;h \to 0[/tex]

As such, ‘x’ is infinitesimally larger than ‘x’. On a graph, ‘x’ lies on the right of x, and such such a limit is called the ‘Right Hand Limit'(RHL). Whereas, if we take, x = c – h, it is called the ‘Left Hand Limit'(LHL). It is denoted as follows:

[tex]RHL:~~~\lim_{x \to c^+} f(x)~~=~~\lim_{h \to 0} f(c + h)[/tex]

[tex]LHL:~~~\lim_{x \to c^-} f(x)~~=~~\lim_{h \to 0} f(c – h)[/tex]

For a continuous function, both these values are equal to each other and also to f(c).

4. List of standard limits

In addition to the identities given in the equation box below, the following standard limits are helpful in evaluating limits of compound functions. The following listed limits are commonly used and are hence enlisted for easy reference:

[tex]\lim_{x \to c} x = c[/tex]

[tex]\lim_{x \to 0} \frac{1}{x} = \infty[/tex]

[tex]\lim_{x \to \infty} x = \infty[/tex]

[tex]\lim_{x \to 0} \frac{\sin(x)}{x} = 1[/tex]

[tex]\lim_{x \to 0} \frac{\tan(x)}{x} = 1[/tex]

[tex]\lim_{x \to 0} \frac{1 – \cos(x)}{x} = 0[/tex]

[tex]\lim_{x \to a} \frac{x^n – a^n}{x – a} = na^{n – 1}[/tex]

[tex]\lim_{x \to a} \frac{\sin(x – a)}{x – a} = 1[/tex]

[tex]\lim_{x \to a} \frac{\tan(x – a)}{x – a} = 1[/tex]

[tex]\lim_{x \to 0} \frac{log_e(1 + x)}{x} = 1[/tex]

[tex]\lim_{x \to 0} \frac{a^x – 1}{x} = log_e a[/tex]

[tex]\lim_{x \to 0} \frac{e^x – 1}{x} = 1[/tex]

 

This has not to be confused with limits of sequences of functions as considered in functional analysis, e.g. example 5.9. in https://www.physicsforums.com/insights/hilbert-spaces-relatives/

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