All four solutions are wrong.

1. Error,

The reason is that the addition of infinite terms cannot be regarded as the addition of finite terms, and the limit cannot be calculated separately.

The correct way is to divide it first.

Original formula = lim [( 1+2+...+n)/n? ]

= Lim [n(n + 1) / (2n？ )]

= Lim [n？ / (2n？ )] + Lim [ 1 / (2n？ )]

= 1/2 + 0

= 1/2

mistake

The reason is that when x →0, sin( 1/x? ) is a bounded quantity, between 1, but notice that its limit does not exist (the last equal sign in your diagram is wrong, although the result is correct)

So x? sin( 1/x？ ) is an infinitesimal multiplied by a bounded variable, and the limit of the whole result is 0.

mistake

The reason is that when x →0, sin( 1/x? Although it is a bounded quantity, it is between 1, but notice that its limit does not exist.

And the denominator is 1/x? → infinity, so sin( 1/x? )/ ( 1/x？ ) Although it conforms to the form of typical limit sinM/M, it does not meet the condition of 1 (it is equal to 1 only when M →0 in the form), but now it is 1/x? → infinity, not →0

As for the correct method, it is the same as the second question, namely

So x? sin( 1/x？ ) is an infinitesimal multiplied by a bounded variable, and the limit of the whole result is 0.

mistake

The whole form is 0/0, and Robida's law can be applied.

Original formula = lim (tanx-sinx)/x = lim (sec? x - cosx) / 1 = 0/ 1 = 0

Or, according to tanx = sinx/cosx.

So,

Original formula = =Lim (tanx-sinx)/x

= Lim (sinx/cosx - sinx) / x

= Lim sinx( 1 - cosx) / (xcosx)

= Lim sinx/x * Lim( 1-cosx)/cosx

= 1 * 0/ 1

= 0

Summary: In fact, advanced mathematics is easier to learn than elementary mathematics. The methods of elementary mathematics are flexible, diverse and complex, while advanced mathematics has many concepts, but the methods are rigid and need a format application. If the application is incorrect, the calculation will be wrong or even impossible.

For example, the calculation of limits has two important limits (it is estimated that teachers and books only say it is important, but why is it important). That is the reality. We spent a whole 1 year studying advanced mathematics, but in the end it was the same as not learning it. Of course, there are many reasons, but in the final analysis, it is rational learning without establishing perceptual knowledge. ........................................................