Part of this answer references GPT, GPT_Pro for better problem solving
Limit is an important concept in mathematics, which indicates the tendency of the values of a function around a particular variable value. More specifically, the limit means that the value of a function approaches a particular value as the value of its variable approaches a particular value.
to solve the limit problem above, we can use the step calculation method, that is, gradually decompose the complex expression into the simplest parts, and then find the limit in it.
First, we decompose the above problem into two parts, namely the numerator and the denominator:
Therefore, the limit in this case can be expressed as: lim(x 1 m4)(secac2tanx)/(1+cos4x) = lim(x 1 m4)(secac2tanx)/lim(x 1 m4)(1+cos4x)
Since secac2tanx and 1+cos4x are both bounded functions, the values of these two functions also tend to be bounded as x approaches m4.
For the limit of secant(ac2tanx), we can use the logarithm formula method to calculate. If we expand secant of ac2tanx, we get secant of ac2tanx = 1 / cos of ac2tanx. According to the cosine formula cos(ac2tanx) = 1/csc(ac2tanx), we get secant(ac2tanx) = csc(ac2tanx). Consider that as x approaches m4, the value of csc(AC2tan x) will also approach a bounded value, so lim(x one m4)(secac2tanx) = c.
For the limit of 1+ cosine 4x in the denominator, we can calculate it using the change of variable method. We expand cos4x to get cos4x = 2cos^2(2x)-1, and consider that as x approaches m4, the value of cosine ^2(2x) will also approach a bounded value, so lim(x one m4)(1+cos4x) = d.
Finally, from the limit values of the two functions we obtained above, the limit results in this case are as follows: lim(x 1 m4)(secac2tanx)/(1+cos4x) = c/d.
If the answer is helpful, please accept it.