Let f (x) be differentiable at x = A and f '(a) not equal to 0. Find the limit of 1 / x power of [f (a + x) / F (a)] when x tends to 0

X → 0lim [f (a + x) / F (a)] ^ (1 / x) = Lim e ^ ln [f (a + x) / F (a)] ^ (1 / x) = e ^ Lim ln [f (a + x) / F (a)] ^ (1 / x) consider Lim ln [f (a + x) / F (a)] ^ (1 / x) = Lim [LNF (a + x) - LNF (a)] / X according to the definition of derivative = [LNF (x)] '| x = a = f' (a) / F (a) so, the original limit = e ^ [f '(a

If f (x) is not equal to 0, for any a and B, f (a + b) = f (a) * f (b), if x1. Find (1) to prove that f (x) is a decreasing function
Find (2) if f (4) = 1 / 16, solve the inequality f (x-3). F (5-x ^ 2)

F (4) = 1 / 16F (4) = f (2) * f (2) = 1 / 16F (2) equals 1 / 4, or - 1 / 4 let a = B, f (2a) = [f (a)] ^ 2 > = 0f (a) and F (2a) have the same sign, f (a) > = 0f (2) = 1 / 4f (x-3) f (5-x ^ 2) = f (x-3 + 5-x ^ 2) = f (- x ^ 2 + X + 2) = 2x ^ 2-x

Find the left and right limit of the function at the specified point, determine whether the limit of the function exists at the point, f (x) = (1 / 2) ^ (- 1 / x ^ 2), x = 0

Both the left and right sides make 1 / X and 178; tend to positive infinity
Then - 1 / X & # tends to negative infinity
So (1 / 2) ^ (- 1 / X & # 178;) tends to be positive infinity
So there is no limit

Find the left and right limit, and determine whether the limit of function at this point exists, f (x) = 3 ^ (1 / 3), x = 0

f(x)=3^(1/3),x=0
This is a horizontal line. Of course, the left and right limits at x0 exist
Left limit = right limit = 3 ^ (1 / 3)

The function f (x) is defined at point X. what is the condition for the existence of limit of F (x) at point x

A function has a limit at a certain point, if and only if the left and right limits exist and are equal. It has nothing to do with whether there is a definition at the point. So there are two cases: 1. At least one of the left and right limits does not exist; 2. Both the left and right limits exist, but are not equal. For example, when f (x) = 1 / x, x approaches 0, the left limit is

What is the relationship between function limit and continuity
Is there a limit for continuity
Whether there is limit is continuous or not

Limit is not necessarily continuous, but continuity must have limit
A continuous function must have two conditions: one is defined here, the other is limited in this interval
Therefore, the limit of a function is a necessary and insufficient condition for the continuity of a function

How to use the continuity of function to find limit
It's better to give some examples

The function f (x) is continuous at x 0. One is that there is a limit and the other is that the limit is equal to the value of the function at x 0
For example:
Let f (x) = xsin 1 / x + A, X

In the limit of function, we have the understanding of lobita theorem and equivalent infinitesimal, the mean value theorem, and the pinch criterion
Don't describe theorems and definitions. What you want is when you think of which theorem to use, what you should pay attention to when you use it, as well as the application practice and differences between several theorems. (don't say to practice more.) I'll also practice more, but I also hope that the experts can give me a brief summary of this kind of questions, methods, and how to get ideas at a glance

Finding the limit of function by lobita theorem
The mean value theorem is used to find the relationship between the change of variable and function, and prove the existence of a certain value
Pinch criterion is used to mark the limit of function size

A new three digit number is obtained by transposing the 100 digits of a three digit ABC with the number on the one digit. The difference between the two numbers is expressed by the formula and the answer is who can divide the difference

The original three digits are: 100A + 10B + C
The new three digits are: 100C + 10B + a
The difference between the two numbers is: 100A + 10B + c-100c-10b-a = 99a-99c = 99 (A-C)
So the difference between the two numbers must be divisible by 99

For a three digit number, the number in the hundreds is a, the number in the tens is B, and the number in the ones is c. the formula for this three digit number is ()
A. abcB. a+b+cC. 100a+10b+c

From the analysis: the three digit is: 100A + 10B + C

Let f (x) be differentiable at x = A and f '(a) not equal to 0. Find the limit of 1 / x power of [f (a + x) / F (a)] when x tends to 0