Find a master to solve a function limit problem

First of all, seeing this way of thinking makes you feel like a good student.

There is an equivalent infinitesimal in the limit of a university. For example, when x→0, sinx and x are equivalent infinitesimal, that is, sinx ~ x.

When x→0, ln( 1+x)~x, in other words, x→ 1, lnx~x- 1

Therefore, it can be substituted, such as x→ 1, lim (x→1) lnx/[2 (x-1)] = lim (x→1) (x-1)/[2 (x).

So what's the problem? It is mentioned in college mathematics that f(x) in limf(x) must be A*B/C, that is, it must be in the form of product or division (as long as one of them is satisfied), so it can be replaced, and your topic is in the form of addition and subtraction, so it cannot be replaced.

Let's use L'H?pital's law to solve this problem:

lim(x→ 1)f(x)

= lim(x→ 1)[xlnx-(x- 1)]/(x- 1)？

= lim(x→ 1)[x(lnx- 1)+ 1]/(x- 1)？

= lim(x→ 1)[x(lnx- 1)+ 1]'/[(x- 1)？ ]' ... using L'H?pital's law

= lim(x→ 1)(lnx- 1+x * 1/x)/2(x- 1)

= lim(x→ 1)(lnx)/2(x- 1)

=lim(x→ 1)( 1/x)/2 ... and then use L'H?pital's law.

= 1/2