Limits are values that we couldn’t reach. We are getting closer to limits but limit can not be reached. For instance if we have: f(x)=(x+1) / (x^2) it is not 0 because f(x)= 1/0 and this will be undefined because when you divide something with zero, the result will be undefined. If there would be a chance for x=0 what would be f(x)? We can find out if we try to find limit of f(x) when x approaches to zero.
Making limits in Calculus tutorial
1) Get a function f(x) = (x – 1) / (x^2 – 1) and in denominator couldn’t be zero. Now, if (x^2 – 1) = 0, for some value of x this means that x can not have the same value. 1^2 – 1 = 0, there are no possibilities for x to be 1. We only can find the limit of f(x) if x goes to one.
2) Finding value for f(x) when:
a) x = 1.5,
b) x = 1.25,
c) x = 1.125
First thing that you will notice is that when you get closer to one, f(x) will get closer to 0.5. You are getting closer to 1 and f(x) is getting closer to 0.5 but you can not reach that values.
3) Factorize the denominator and this will help you to find limit by factoring. Now you should use your knowledge from algebra classes: (A + B)(A – B) = (A^2 – B^2). Now you have (x^2 – 1) = (x + 1)(x – 1). When you apply this you will have f(x) = [(x - 1) / (x + 1)(x - 1)].
4) Now you will have to find the slope: Dy/Dx. These are two points in a line and the that with a higher value of x is called Point 2. The point with the x which has lower value is called Point 1. Dy = (y2 – y1) and y2 value is y on a Point 2 and y1 is that on a point 1. Dx will be equal to (x2-x1) and x2 is on a point 2 and x1 is that on a point 1.
5) Find the slope of f(x) = x^2. x1 =1 and x2 = 3, and this is one slope. If x1 = 1 and x2 = 2 these will be different slopes. If f(x) is parabola, the slope will be in change, which means that function will have different slopes on different points. If you want to find a slope on certain point you can do a limit at that point if you use Dy / Dx. You should make Point 2 as much closer as it is possible to Point 1. The same rule you can do for Point 1 if x value is 1 or 2 or 3.
6) When you find values, write them for the limit of Dy/Dx for f(x) = x^2: 14 when x = 7, 12 when x = 6, 10 when x = 5, and 8 when x = 4. When you do this, you will find the limit of Dy/Dx when x is 1,2 or 3. This will make slope of f(x) at x point = x2. These are the derivatives of functions.
7) You can apply the same rule from tip 5 to find Dy/Dx for functions: f(x) = 2x^2, g(x) = x^3, and h(x) = 2x^2 + x^3. When you find the limit, you will have the derivative.