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Trigonometric limits involve evaluating limits that contain trigonometric functions like \( \sin \), \( \cos \), or \( \tan \). Understanding how these functions behave as they approach certain values is crucial. When dealing with trigonometric limits, especially those approaching zero, knowing the fundamental values of sine and cosine is helpful.
For example, both \( \sin(0) \) and \( \cos(0) \) give clear values: \( \sin(0) = 0 \) and \( \cos(0) = 1 \). These foundational values allow us to directly solve limits without complications.
Trigonometric limits often require special techniques such as using identities or L'Hôpital's Rule, but in simpler cases like the exercise above, a direct evaluation gives the correct answer. Staying familiar with the unit circle and the values of trigonometric functions at key angles (like 0, \( \pi/2 \), \( \pi \), etc.) can simplify solving these limits.

Direct substitution is a technique where you simply replace the variable in the function with the value it approaches. This method is straightforward and works well when the limit leads to a determinate form.
In the provided problem, when we substitute \( t = 0 \) into \( \frac{\cos^2 t}{1+\sin t} \), the expression becomes \( \frac{1}{1} \) which equals 1. Since this does not result in an undefined situation like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), direct substitution is valid here.
For direct substitution to work, ensure the expression isn't leading to an indeterminate form. When it's applicable, it's a reliable and easy technique to find limits, saving time and effort compared to more complex methods.

Indeterminate forms occur when direct substitution in a limit problem leads to expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), or others that do not clearly define a limit. These forms signal that further algebraic manipulation or advanced techniques are necessary to find the limit.
In our exercise, substituting \( t = 0 \) into the expression results in \( \frac{1}{1} \), which is a determinate form, not indeterminate. This means we do not need additional manipulation to solve the problem and can conclude the limit is simply 1.
Recognizing when an expression is in an indeterminate form is critical. It guides you to apply other strategies like factoring, using conjugates, or L'Hôpital's Rule to resolve these limits. Being familiar with different indeterminate scenarios helps in choosing the right strategy when direct substitution doesn't work.