91Ó°ÊÓ

Trigonometric limits play a vital role when dealing with calculus problems, especially those involving trigonometric functions like sine, cosine, and tangent. Understanding these limits can help us evaluate expressions as the input gets closer to a particular value. The limit

• helps in approximating complex functions near a point
• is essential for finding derivatives
• supports the definition of continuity.

For the problem involving \( \lim_{x \rightarrow 1 / 2} x^{2} \tan(\pi x) \), knowing the behavior of the tangent function is critical.
The tangent function \( \tan(\theta) \) behaves peculiarly at certain points, such as \( \pi/2 \), where it becomes undefined due to vertical asymptotes. Recognizing these key characteristics helps us predict when standard limit rules can be applied or when alternative strategies need to be employed.

Limit substitution is an essential technique in calculus for simplifying expressions before evaluating the limit. By substituting a specific value for \( x \), you attempt to directly compute the limit. In the example, \( x \) is replaced with \( 1/2 \), which transforms the expression to \( (1/2)^2 \tan(\pi/2) \).
Using substitution can help determine if the limit can be directly calculated.

• If after substitution the function is still defined, you can compute the limit.
• If substitution leads to an undefined expression, further analysis might be required.

In this case, because \( \tan(\pi/2) \) is undefined, it indicates a potential issue. Thus, substitution serves the crucial role of identifying problematic points in the function.

Exploring limits at undefined points requires careful consideration, as these points often signal where a function's behavior changes drastically. In the function \( x^{2} \tan(\pi x) \), the substitution showed \( \tan(\pi/2) \) being undefined.

• Undefined points usually indicate a vertical asymptote.
• A function's limit might not exist at these points without further mathematical manipulation.

In our problem, the limit does not exist because, at \( x = 1/2 \), the tangent function reaches an asymptote, making the overall expression misbehave.
These undefined points can often be handled by other techniques, such as rearranging the expression or using L'Hôpital's rule—but in some cases, like this one, acknowledging the non-existence of the limit is necessary.