Limit of a function

Calculation of `lim_(x -> x_0) f(x)\text{ or }lim_(x -> x_0^+) f(x)\text{ or }lim_(x -> x_0^-) f(x)\text{ or }lim_(x -> -oo) f(x)\text{ or }lim_(x -> +oo) f(x)`
input a single letter
To multiply: write a*b not ab
+inf (for +infinite) or -inf (for -infinity)

This tool is designed to calculate the limit of a function at a specific point. You have a variety of options for setting the x-value that the function approaches. These options include:

  • A numerical value, like 0.
  • A constant such as pi or e.
  • An expression combining numbers and constants.
  • Positive infinity, which you can enter as +inf.
  • Negative infinity, entered as -inf.

In addition, you have the flexibility to determine the limit from a specific direction. If you want to calculate the right-hand limit, where x approaches x0 from the right, select '+' in the direction field. For the left-hand limit, choose '-'.

The calculator supports a wide range of common functions, including sine, cosine, tangent, logarithm (log), exponential, and square root, among others.

How to use this calculator ?

Variables A function can have one or more variables, but only one main variable.
A variable is a single lowercase or uppercase letter.
A function f with one main variable : f(x) = 4*x
A function g with one main variable x and a secondary parameter m,
g(x) = 4*x*m + x + 1,
In this case, enter x in the “main variable” field
Numbers Use a dot as decimal separator
Operators + (addition),
- (substration),
* (multiplication),
/ (division),
^ (power),
For multiply operator, enter a*b not a.b nor ab. Example: 2*x.
Constants You may use these constants :
pi (approx. 3.14),
e (approx. 2,72)
Examples: f(x) = pi * x or f(x) = e * (x+1+2*e)2
Common Functions You may use theses functions in the expression of f(x)
sqrt(x) (square root),
exp(x) (exponential function),
log(x) or ln (natural logarithm),
Trigonometric functions You may use theses functions in the expression of f(x)
sin (sine),
cos (cosine),
tan (tangent),
cot (cotangent),
sec (secant),
csc (cosecant),
Inverse trigonometric functions You may use theses functions in the expression of f(x)
arcsin (arcsine),
arccos (arccosine),
arctan (arctangent),
arccot (arcotangent),
arcsec (arcsecant),
arccsc (arccosecant),
Hyperbolic Functions You may use theses functions in the expression of f(x)
sinh (hyperbolic sine),
cosh (hyperbolic cosine),
tanh (hyperbolic tangent),
coth (hyperbolic cotangent),
sech (hyperbolic secante),
csch (hyperbolic cosecant)
Inverse hyperbolic functions You may use theses functions in the expression of f(x)
asinh (inverse hyperbolic sine),
acosh (inverse hyperbolic cosine),
atanh (inverse hyperbolic tangent),
acoth (inverse hyperbolic cotangent),
asech (inverse hyperbolic secant),
acsch (inverse hyperbolic cosecant)

What is the Purpose of a Function Limit?

The limit of a function at a given point tells us about the behavior of this function as the value of x approaches this point, without reaching it.

Notation: If the limit of `f(x)` as x approaches a, where a and L are real numbers, is equal to L, then we denote it as `lim_(x -> a) f(x) = L`. This means that as x becomes very close to a, the value of the function `f(x)` becomes very close to L.

Important Remark: the function must be defined in the neighborhood of a, but not necessarily at a!

The definition and notation remain valid even if "a" and/or "L" are replaced by plus or minus infinity. For example, `lim_(x -> +oo) f(x) = L` means that as x becomes very large, the value of the function `f(x)` approaches L.

Left-hand and Right-hand Limits

There are two ways for x to approach a:

  • From the right: x approaches a from the right (noted as `x -> a+`), meaning x gets close to a while remaining greater than a.
  • From the left: x approaches a from the left (noted as `x -> a-`), meaning x gets close to a while remaining less than a.

This distinction is crucial because for some functions, the right-hand limit can be different from the left-hand limit. For example, `lim_(x -> 0+) 1/x = +oo` but `lim_(x -> 0-) 1/x = -oo`. If the two limits are not equal, the limit at this point is not defined.

How to Calculate a Function's Limit?

Direct Substitution Method (case `x -> a`)

Direct substitution is the first technique to try. It involves replacing x with a to examine the behavior of the function near a.

Direct substitution can lead to a determined form or an indeterminate form. In the latter case, it is necessary to resolve the indeterminacy by applying techniques such as simplification or multiplying by the conjugate.

List of Defined (or Determined) Forms
We note:
p (as in positive) a nonzero positive real number,
n (as in negative) a nonzero negative real number,
q (a number of any qualconque sign, positive or negative non-zero),
`+oo`, plus infinity,
`-oo`, minus infinity,
`oo`, plus or minus infinity.

Defined Forms: Addition and Subtraction
`+oo + +oo = +oo` `-oo - -oo = -oo`
`q + (+oo) = +oo` `q + (-oo) = -oo`
Defined Forms: Multiplication and Division
`+oo * +oo = +oo` `-oo * -oo = +oo`
`p * (+oo) = +oo` `n * (+oo) = -oo`
`p * (-oo) = -oo` `n * (-oo) = +oo`
`(+oo) / p = +oo` `(+oo) / n = -oo`
`(-oo) / p = -oo` `(-oo) / n = +oo`
`q / oo = 0` `0 / oo = 0`
Defined Forms: Power
`0^(+oo) = 0` `0^(-oo) = oo`
List of Indeterminate Forms

`+oo - oo`
`0 * oo`

Among the most commonly used techniques to resolve indeterminacy are factoring, applying a notable identity, and applying L'Hôpital's rule: `lim_(x -> a) f(x)/g(x) = lim_(x -> a) frac{f'(x)}{g'(x)}`.

Calculating a Limit as `x -> oo`

When x tends towards infinity, replace x with very large values to understand the behavior of the function. For example, `lim_(x -> +oo) 1/x^n = 0` for a positive integer n.
`lim_(x -> -oo) 1/x^n = 0`,
`lim_(x -> +oo) x^n = +oo`,
`lim_(x -> -oo) x^n = +oo`, if n is even
`lim_(x -> -oo) x^n = -oo`, if n is odd

Simple Examples of Limit Calculation or with a Determined Form

Example 1: A simple case of direct substitution

`lim_(x -> 0) (x^2+1) = 1`
As `x` approaches closer to 0, then `x^2` also approaches 0, so `x^2+1` approaches 1.

Example 2: A case of a determined form (`1/(0+)`)

`lim_(x -> 0) 1/x^2 = +oo`
As `x` approaches closer to 0, `x^2` also approaches 0 (from the positive side as `x^2 >= 0`), so `1/x^2` becomes increasingly large (try calculating `1/0.0000001^2`!) which means it tends towards `+oo`.

Example 3: A case of a determined form (`1/(+oo)`)

`lim_(x -> +oo) 1/(x+1) = 0`,
As `x` becomes larger and larger, `1/(x+1)` becomes smaller and approaches 0.

Examples of Limit Calculation with an Indeterminate Form

Example A: Indeterminate form `0/0`

Calculate `lim_(x -> 1) (x-1)/(sqrt(x)-1)`

The function in question is well defined in the neighborhood of 1 (but not for 1) because the denominator must be different from 0. Additionally, x must be positive (due to the square root). The domain of definition is the set of positive real numbers, excluding 1.

Direct substitution gives `0/0`, but we observe that `x = (sqrt(x))^2`, so we can apply a remarkable identity:

`f(x) = ((sqrt(x))^2 -1 ) / (sqrt(x) - 1) = ( ((sqrt(x) -1 ) * (sqrt(x) +1 )) / (sqrt(x) - 1)) = sqrt(x) +1`

By performing a direct substitution, it is easily deduced that

`lim_(x -> 1) (x-1)/(sqrt(x)-1) = 2`

See also

Derivative of a function
Primitive calculator
Taylor series expansion
Value of a function
Definite Integral