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How can we learn the limits of function and Pythagorean theorem easily?

Pythagorean Theorem:

The Pythagorean Theorem was one of the earliest theorems of ancient civilizations. This famous theorem had named after the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students. 

The Pythagorean Theorem is a statement about triangles containing a right angle. This theorem is named after the mathematician Pythagoras. It comprises three sides and is applied on the sides of the triangle only. These sides are named as the base of the triangle, the perpendicular of the triangle and the hypotenuse of the triangle. The base and perpendicular have an angle of 90 degrees. This theorem states that the sum of the squares of the base and perpendicular equals the square of the hypotenuse. This theorem is only applicable to the right-angled triangles. 

right-angle-triangle

There are three sides to the triangle. The side “a” will be the perpendicular of the triangle. Also, The side “b” will be the base of the triangle and the side “c” is the hypotenuse of the triangle.  According to the theorem, the sum of the squares of the base and perpendicular equals the square of the hypotenuse.

c2= a2+b2

Hint to remember the sides of base and perpendicular:

These two sides will always have a mutual angle of 90 degrees as shown in the figure. 

Example to solve a problem:

Suppose we have given the values of hypotenuse as 13 and the value of base as 5. We need to calculate the value of perpendicular. So we will put the values in the given formula. The thing we need to memorize the directions and the formula so that we can calculate the value of the other side.

c2= a2+b2

132= 52+ b2

169= 25+ b2

169-25= b2

144= b2

Now, taking the square root on both sides.

144= b2

b =12

So, the value of the hypotenuse comes out to be 12. 

Let us consider another example of the Pythagorean theorem. The value of perpendicular and hypotenuse are 20 and 25. The value of the base is not available and has to calculate via the Pythagorean theorem. 

c2= a2+b2

Here, a and b are base and perpendicular respectively and c represents the hypotenuse. Now by putting the values in the above equation we get

252= a2+202

625= a2+400

625-400 = a2

a2= 225

a= 15

Hence, the value of the base is 15. It should be noted here the value of either parameter that is calculated via Pythagorean theorem can be cross-checked by again putting the values in the given formula or by using an online Pythagorean calculator

Limits of the function:

The limit of a function at a point a in its domain (if it exists) is the value that the function approaches as its argument approaches a. The concept of a limit is the fundamental concept of calculus and analysis. It uses to define the derivative and the definite integral, and it can also use to analyze the local behavior of functions near points of interest.

limit

To evaluate lim x→a f(x), we begin by completing a table of functional values. We should choose two sets of x -values—one set of values approaching “a” and less than “a”, and another set of values approaching “a” and greater than “a”. 

One-sided limit:

Let Lim x →a − o denote the limit as x goes toward a by taking on values of x such that x < a. The corresponding limit lim x →a−0 f (x) is the left-hand limit of f (x) at the point x=a.

Similarly, let lim x→a+0 denote the limit as x goes toward a by taking on values of x such that x >a. The corresponding limit limx→a+0 f(x) is the right-hand limit of f(x) at x=a.

Note that the 2-sided limit Lim x→a f(x) exists only if both one-sided limits exist and are equal to each other, that is lim x→a−0 f(x) = limx→a+0 f(x). In this case, lim x→a f (x)=limx→a−0 f(x)= Lim x→a+0 f(x).

Example 1:

Prove that: 

example 1

We assume that ε>0.  Find a number N such that for any x > N the following inequality is valid:

eg 1

Convert this inequality to get:

eg 1.1

Since 0<x< N, then x+1>0, so we can simply write

eg 1 solution

Example 2:

example 2

Solution:

solution 2

Limits at infinity for rational functions:

There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):

  • If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
  • Also, If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q.
  • Moreover, the limit is 0, If the degree of p is less than the degree of q.

When limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

Hence, it can be concluded from the above study that the easiest way to learn the Pythagorean formula and the limits calculation is by solving their mathematical problems daily.

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