Limit problems with trigonometric functions are a very common class of problems in mathematics, and such problems usually involve the periodicity, boundedness, and some specific trigonometric identities of trigonometric functions. Solving this type of problem usually requires a number of skills, including splitting the limit with periodicity, using the range of values of bounded control terms, and simplifying with specific trigonometric identities.
Finding the limit of trigonometric functions requires the following considerations:
1.Periodicity: Many trigonometric functions (e.g., sine, cosine) have a definite period, which helps us to split the limit, or to control the value of the function within a certain range.
2.Bounded: Trigonometric functions are bounded, which means that the value of a function will not be unbounded in any finite interval. This provides an important constraint for us to deal with limit problems with trigonometric functions.
3.Handling of specific points: In some cases, a function may affect the behavior of the limit at certain specific points (e.g., extreme points, non-derivable points). Particular attention needs to be paid to these points when dealing with such issues.
4.Equivalent Infinitesimal Substitutions: When dealing with limit problems with trigonometric functions, you can sometimes use equivalent infinitesimal substitutions to simplify expressions. But care must be taken when using this substitution, as not all infinitesimal can be replaced at will.
5.*Utilizing Taylor Expansion**: For some complex trigonometric expressions, Taylor Expansion can be an effective tool. With Taylor expansion, we can represent complex functions in the form of the sum and product of polynomials, which helps us better understand and deal with limits.
Let's take a few concrete examples to illustrate how to solve a limit problem with trigonometric functions
Example 1**: Find $ lim frac$
Analysis: This is a fundamental limit problem that can be solved by using equivalent infinitesimal substitutions. We know that when $x to 0$, $sin x sim x$.
Answer**: Based on the equivalent infinitesimal substitution, we have $ lim frac = lim frac = 1$.
Example 2**: Find $ lim frac$
Analysis**: Since $ sin x$ can be taken in the range of $[1, 1]$ and $x 2$ is unbounded, this limit does not exist.
Answer**: Since $ sin x$ can be taken in the range of $[1, 1]$ and $x 2$ is unbounded, this limit does not exist.
Example 3**: Find $ lim frac$
Analysis: This limit involves two complex functions and a square term. We can try to simplify this expression using Taylor expansion.
Answer**: First, we know that $ cos x = 1 - frac + o(x 4)$ and $e x = 1 + x + o(x 2)$. Substituting these Taylor expansions into the original limit yields $ lim frac + o(x 4) +1 + x + o(x 2)}$ to get $ lim frac = - frac$.
Example 4**: Find $ lim frac$
Analysis**: This limit involves two different trigonometric functions and a cubic term. We can try to simplify this expression using equivalent infinitesimal substitutions and trigonometric properties.
Answer**: First, we know that $ sin(x 2) = x 2 + o(x 4)$ and $ sin(2x) = 2x + o(x 3)$. Substituting these into the original limit gets: $ lim frac$. Simplify to $ lim frac = - frac$.
Through the above examples, we can see that solving limit problems with trigonometric functions requires a combination of mathematical tools and techniques, including equivalent infinitesimal substitutions, Taylor expansions, periodicity, and boundedness. At the same time, we should also pay attention to the handling of some special cases, such as the treatment of specific points and the judgment of infinity or infinitesimal and so on.