Welcome to our online limit calculator, which is designed to help you in solving calculus problems related to limits. A limit is defined as a vital tool in calculus, used to describe whether a sequence or function approaches a stable (fixed) value as its index or input reach a set point.

These can be defined for distinct series, as functions of one or more real-valued inputs or complex-valued operations. Don’t worry! Our multivariable limit calculator can handle it. In this article, we will describe; what is the limit of a function? The step by step calculation, and applications in daily life.

## What is the limit of a function?

To explain it, let’s suppose *f* as a real valued function and b as a continuous quantity (a real number).

Instinctively speaking, equation would be as follows:

$$ \lim_{x\to\ b} f \left( x \right) = \text{L} $$

This illustrates that *f*(x) can be set as near to L as preferred by making x suitably close to b. In that case, the above expression can be defined as the limit of the function f of x, as x approaches b, is equal to L.

**Example:**for x=1, x^{2}-1/x-1 = 1^{2}-1/ 1-1 = 0/0 now, this is undefined or indeterminate, we need another way to work this out.

Now, instead of x=1, this time, we will try approaching it a little bit closer:

x | (x^{2} − 1)/(x − 1) |
---|---|

0.25 | 1.0625 |

0.45 | 1.2025 |

0.9 | 1.810 |

0.99 | 1.99000 |

0.999 | 1.99900 |

0.9999 | 1.99990 |

Now, we have witnessed, as x gets close to 1, the other function gets closer to 2, and so we can express it as:

$$ \lim_{x\to\ 1} \frac {x^2-1} {x-1} = 2 $$

For any chosen degree of nearness ε, one can determine an interval nearby x_{0}(or previously assumed b). Because, the given values of *f(x)* computed here varies from L by a quantity less than ε (i.e., if ε= |x − x^{0}| < δ, then |f (x) − L| < ε). This can be used to determine whether a given number is a limit or not.

The estimation of limits, particularly of quotients, typically involves adjustments of the function in order to write it in a more obvious form, as shown in the above example.

Limits are used to calculate the rate of change of a function, and as approximations, throughout the analysis to get to the nearest possible value. For example, an area inside a curved region, may be described as limits of close estimations by rectangles.

## How to calculate limits?

There are a range of techniques used to compute the limits, we will discuss some ways to algebraically calculate these values:

## By including the x value:

This method is simple, all you need to do is plug in the value of x that is being approached. If you get a 0 (undefined value) move on to the next method. But if you get a value it means your function is continuous, and you’ve acquired the desired result.

Example:Find

$$ \lim_{x\to\ 5} \frac{x^2-4x+8} {x-4} $$

Now, put the value of x in equation = $$ \frac{5^2- 4*5 + 8}{5-4} =\frac{25-12}{1} = 13 $$

## By Factoring:

If the first method fails, you can try factorization technique, especially in problems involving polynomial expressions. In this method, we first simplify the equation by factoring, then cancel out the like terms, before introducing x.

Example:Find

$$ \lim_{x\to\ 4} \frac{x^2-6x-7} {x^2-3x-28} $$

Now,factorize the equation $$=\;\frac{(x-7)(x+1)}{(x+4) (x-7)}$$

Here, x-7 will cancel out, the next step is to put the x value $$=\;\frac{(4+1)} {(4+4)}\;=\;\frac{5}{8}$$

## By rationalizing the numerator:

The functions having square root in the numerator and a polynomial expression in the denominator, requires you to rationalize the numerator.

**Example:** Consider a function, where x approaches 13:

$$g(x)=\frac{\sqrt{x-4}-3}{x-13}$$

Here, x inclusion fails, because we get a 0 in the denominator and factoring fails as we have no polynomial to factorize. In this case we will multiply both numerator and denominator with a conjugate.

**Step 1:** Multiply conjugate on top and bottom.

Conjugate of our numerator: $$\sqrt{x-4}+3$$

$$\frac{\sqrt{x-4}-3}{x-13}.\frac{\sqrt{x-4}+3}{\sqrt{x-4}+3}$$

$$(x-4)+3\sqrt{x-4}-3\sqrt{x-4}-9$$

**Step.2:** Cancel out. Now it will be further simplified to x-13 by cancelling the middle alike terms. After cancelling out:

$$\frac{x-13}{(x-13)(\sqrt{x-4}+3)}$$

Now, cancel out x-13 from top and bottom, leaving:

$$\frac{1}{\sqrt{x-4}+3)}$$

Step 3: Now after incorporating 13 in this simplified equation, we get the results 1/6.

Quite lengthy and time consuming process isn’t it? Well no problem, with the use of our smart **central limit theorem calculator**, you will get the desired value in seconds.

The use of our **limit calculator with steps:**

**Step 1:** Input the required function

**Step 2:** Enter the value to approach, then press compute, that’s it leave the math work on our gizmo. You will get the limit within seconds.

## Applications in daily life:

Real-life limits can be seen in a broad range of fields. For instance, the quantity of the new compound derived from a chemical reaction can be considered as the limit of a function as time reaches infinity. Likewise, measuring the temperature of an ice cube placed in a warm glass of water is also a limit.

These are also applied in real-life estimates to compute derivatives. It is quite complex to approximate a derivative of complicated motions. The engineers uses small differences in function to approximate derivative. Other areas involves the estimation of social security income limits. Use our social security earnings limit calculator for this purpose.

We are optimistic, that this article will benefit you in understanding and applying the concepts of this important calculus tool. Best of luck!