Limit Calculator Online
--
Before I talk about calculating limits with uncertainty, I want to believe that you already have an understanding of what a limit is and how to calculate elementary limits. If there is no such understanding, first read the article “Limits. The concept of limits. Calculation of limits.”
We now turn to the consideration of limits with uncertainty.
There is a group of limits, when x is the Arrow of Infinity, and the function is a fraction, substituting in which the value x = Infinity will receive the uncertainty of the form Limits with uncertainty.
Example Limit Calculator Online
It is necessary to calculate the limit. Limits with uncertainty.
We use our rule №1 and substitute the Infinity in the function. As you can see, we get uncertainty. Limits with uncertainty.
In the numerator we find x in the highest degree, which in our case = 2:
Limits to uncertainty
Do the same with the denominator:
Limits to uncertainty
Here also the highest power = 2.
Next, you need to choose the highest of the two degrees found. In our case, the degree of the numerator and denominator coincide and = 2.
So, to uncover uncertainty Limits with uncertainty, we will need to divide the numerator and denominator by x to the highest degree, i.e. on x2:
Limits to uncertainty
Answer: 2/3.
There are also limits with another uncertainty — the type Limits with uncertainty. The difference from the previous case is that x tends not to infinity, but to a finite number.
Example.
It is necessary to calculate the limit limits with uncertainty.
Again, use rule number 1 and substitute the number -1 in place x:
Limits to uncertainty
We have obtained uncertainty Limits with uncertainty, for the disclosure of which it is necessary to factor the numerator and denominator, which in turn usually solves a quadratic equation or use abbreviated multiplication formulas.
In our case, we solve the equation:
Limits to uncertainty
Find the discriminant:
Limits to uncertainty
Limits to uncertainty.
If the root is not extracted the whole is most likely D calculated incorrectly.
Now we find the roots of the equation:
Limits to uncertainty
Limits to uncertainty
Substitute:
Limits to uncertainty
Numerator laid out.
In the denominator, we have x + 1, which is therefore the simplest factor.
Then our limit will look like:
Limits to uncertainty
x + 1 beautifully reduced:
Limits to uncertainty
Now we substitute the value -1 in the function instead of x and get:
2 * (- 1) — 5 = -2–5 = -7
Answer: -7.
Consider the basic provisions used in solving various kinds of problems with the limits:
The limit of the sum of 2 or more functions is equal to the sum of the limits of these functions:
Limits — Rules
The limit of a constant value is the most constant value:
Limits — Rules
For the sign of the limit you can make a constant coefficient:
Limits — Rules
The limit of the product of 2 or more functions is equal to the product of the limits of these functions (the latter must exist):
Limits — Rules
The limit of the ratio of 2 functions is equal to the ratio of the limits of these functions (in that case, if the limit of the denominator is not equal to 0:
Limits — Rules
The degree of the function under the limit sign applies to the very limit of this function (the degree must be a real number):
Limits — Rules
On this with the calculation of limits with uncertainty everything. Even in the article “Remarkable Limits: The First and Second Remarkable Limits” we separately consider an interesting group of limits. The article will insert another block to address most of the limits found in the vast learning space.
