Solution idea
A, because (x? + 1)/(3x？ -2) is a continuous function, so the limit can directly substitute x=0 into the limit function to get its limit value.
B, for a radical function, we must first calculate the radical in a rational way, convert the subtracted radical into the added radical, and then calculate the limit value.
C, because the reciprocal of e (-x) is1/e (-x), when x→∞, its value is equal to zero.
D, when x→∞, 1/x? Is infinitesimal, and |cos(2x- 1)|≤ 1, so its limit value is equal to zero.
Solution process
Topic knowledge point
1, continuous function. A continuous function refers to the function y=f(x). When the change of independent variable X is small, the change of dependent variable Y is also small. For example, the temperature changes with time, as long as the time change is small, the temperature change is small; For another example, the displacement of a free-falling body changes with time. As long as the time change is short enough, the displacement change is also small. For this phenomenon, the dependent variable changes continuously about the independent variable, and the image of continuous function in rectangular coordinate system is a continuous curve without fracture. According to the nature of limit, the necessary and sufficient condition for a function to be continuous at a certain point is that it is continuous near that point.
Definition of continuous function:
2. Roots are physical and chemical. In mathematics, the rationalization of roots is an important skill, which can help us simplify roots into a form that is easier to calculate. Radical rationalization is actually simplified by using the square difference formula. That is, (a+b)(a-b)=a? -B?
3. Infinitely small. Infinitesimal is a concept in mathematical analysis. In classical calculus or mathematical analysis, infinitesimal usually appears in the form of functions and sequences. ? [1] Infinitely small quantity is a variable with a limit of 0, which is infinitely close to 0. Specifically, when the independent variable x is infinitely close to x0 (or the absolute value of x is infinitely increased), the function value f(x) is infinitely close to 0, that is, f(x)→0 (or f(x)=0), then f(x) is called the infinitesimal amount when x→x0 (or x→∞). In particular, we should not confuse very small numbers with infinitesimal numbers.
Definition of infinitesimal:
? Infinitesimal property:
1, infinitesimal is not a number, but a variable.
2. Zero can be used as the only constant of infinitesimal quantity.
3. Infinitely small quantities are related to the trend of independent variables.
4. If the function g(x) is bounded in the hollow neighborhood of x0, it is called bounded quantity when x→x0.
5. The sum of finite infinitesimals is still infinitesimal.
6. The product of finite infinitesimals is still infinitesimal.
7. The product of bounded function and infinitesimal is infinitesimal.
8. Especially the product of a constant and an infinitesimal is also an infinitesimal.
9. The reciprocal of infinitesimal with non-zero constant is infinity, and the reciprocal of infinity is infinitesimal.