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Significance of limit functions in math

How much important it is to learn limit function and what is its significance in math



What does it mean by limit?

The concept of limit or limiting process is not new in calculus the term limit” has been around for a thousand years. The earlier mathematicians were formed on using limits while dealing with “circles” it helps in deriving better approximations of the circle. The modern definition of limit came in the late 19th century. Hence following the footsteps of our mathematical ancestors we have maintained the quest of learning and applying “limit” in calculus. 

A limit shows us whether a function or a sequence approaches a “stable” or fixed value when its index goes to a set point! 

What is the limit of a function?

When we have talked about a limit of a function it simply throws light on the fact that “function has a limit at a given or specific point” 

To understand the concept more effectively, let’s consider a function f(X) in which f is a real-valued function whereas b as a continuous quantity. i.e

limx bf(x)=L

The equation shows that the function f(x) can be set close to L if we have preferred arranging the value of x close to b. And it will be defined as the limit of function f(x) f of x, as x approaches b, equal to L.

For instance, x=1, x2-1/x-1 = 12-1/ 1-1 = 0/0

The above expression is indeterminate to make it work. We will arrange the value of x very close to 1 and derive a suitable expression for it. For example, 

We can set the value of x to 0.25 and the resultant in x2-1/x-1 will be 1.0625 similarly if we have arranged the value of x to 0.45 the final value will be 1.205. For x=0.9 the function will be 1.810.

If you have noticed if we have arranged the value of x closer to 1 the other functions approaching closer to 2. We will write the equation like:

limx→ 1 x2−1 / x−1 = 2

From the above example, the concept of the limit will be clear that it measures the rate of change of function by the approximations to get the nearest possible value. 

This is the most basic step where we have calculated limits. Moving to the next steps the complexity will increase. You can calculate limits by putting the formula or you can use the limit calculator which will simplify the steps for you and derive accurate results in a split second.

What is the quadratic formula?

The quadratic formula is commonly used in mathematics. it is used to solve the polynomial equations like ax2+ bx + c = 0 in this polynomial equation “a” represents quadratic coefficient “b” is the linear coefficient and “c” is constant. Here the value which has to be found is “x”. We cannot set the value of “a” to zero. If we do so then it must lie in linear form. 

There are many ways of solving quadratic equations. Apart from the quadratic formula, we can solve the quadratic equations by factorization, completing the square, or the graphical methods. Solving a quadratic equation also called roots. And typically one equation carries one or two roots. It means that there are one or two solutions to the equation. 

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We have to extract the standard form of a quadratic equation

As stated above that the ideal quadratic equation looks like

ax2 +bx +c = 0

But sometimes the equation is not present in its original form. Rather it is hidden or complicated but it is the basic step to extract the ideal form of a quadratic equation

For example, the equation may be written as x2 = 3x – 1, in this case, we have to shift all the terms to the left side and it will be x2 -3x +1=0

Let’s take another example, in the equation 2(w2 – 2w) = 5 we will solve this equation first by multiplying 2 inside the bracket 2w2 – 4w = 5 and now we shift the term to left side 2w2 – 4w – 5 = 0  

After simplifying the quadratic equation into simpler parts we will apply the quadratic formula:

The Discriminant | Digestible Notes

Apart from this formula, the other ways like completing the square and graphical methods are used to solve the quadratic equation. Simpler quadratic equations are easy to solve but as we move on the complexity increases. If you want to avoid the manual calculation you can use the quadratic formula calculator it will help you to derive accurate results without any error.

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