How to judge whether a function has limit, continuity, differentiability and differentiability?

A function can be derived as long as its image is continuous, and its differentiability should be holographic continuous. Continuity depends on the domain (if it is continuous in high school), and the limit requires continuity, which depends on the range of the function. The range of a function must be meaningful at one end, that is, it cannot be infinite, and the domain should be infinite at this end, so that the function has a limit at this end.

When the denominator is equal to zero, the trend value cannot be directly substituted into the denominator, which can be solved by the following small methods:

First: factorization, which makes the denominator not zero through simplification.

Second: If there is a root sign in the denominator, you can match a factor to remove the root sign.

Third: the solutions I mentioned above are all carried out under the condition that the trend value is fixed. If it tends to infinity, the numerator and denominator can be divided by the highest power of the independent variable. This theorem is usually used: the reciprocal of infinity is infinitesimal.

Extended data:

A real variable function is a differentiable function if its derivative exists at every point in the domain. If f is a differentiable function at x0, then f must be continuous at x0. In particular, any differentiable function must be continuous at every point in its domain. The reverse is not necessarily true. In fact, there is a function that is continuous everywhere in its domain, but it is not differentiable everywhere.

What if? Then it is differentiable at X0. At this point it must be continuous. In particular, all differentiable functions must be continuous at any point in their domain. The converse proposition does not hold: continuous functions may not be differentiable. For example, a function with vertices, cusps, or vertical tangents may be continuous, but it is not differentiable at abnormal points.

If all partial derivatives of a function exist and are continuous in the neighborhood of a point, then the function is differentiable at that point and in the form of Class C. (this is a necessary and sufficient condition for differentiability), and a multivariate real function f:R→R is differentiable at point x0.

Baidu Encyclopedia-Function Limitation