91Ó°ÊÓ

Understanding limits is a fundamental aspect of calculus, and the direct substitution method is often the first tool utilized when evaluating limits. This approach is quite straightforward—when trying to find the limit of a function as the variable approaches a particular value, simply substitute that value into the function.

However, direct substitution is not always applicable. It only works when the function is continuous at the point of interest. If the function is not continuous, direct substitution may lead to undefined expressions, such as dividing by zero, or other indeterminate forms. In such cases, alternative methods such as factoring, rationalization, or L'Hôpital's rule might be used.

In the given exercise, the direct substitution method is applied successfully because the function \( \sec \frac{\pi}{2^{x}} \ln x \) does not lead to any undefined expressions when \( x = 1 \) is substituted.

The natural logarithm, denoted by \(\ln(x)\), is a fundamental mathematical function that plays a crucial role in various branches of mathematics, particularly calculus. It is the inverse of the exponential function \( e^x \) where \( e \) is Euler's number, approximately equal to 2.71828. The natural logarithm has a domain of all positive real numbers, and it measures the time needed to reach a certain level of continuous growth.

Understanding the properties of the natural logarithm is essential for students. Here are a few key properties:

• \( \ln(1) = 0 \) - The logarithm of 1 is always zero, because \( e^0 = 1 \) by definition.
• \( \ln(e) = 1 \) - The natural log of \( e \) is 1, as \( e^1 = e \) by definition.
• The function \( \ln(x) \) is continuous and differentiable for all \( x > 0 \).

This forms the basis of step 2 in the solution, where the natural logarithm of 1 is correctly evaluated to be 0.

Trigonometric functions are a vital component of mathematics, providing a relationship between the angles and lengths of triangles. In calculus, these functions, which include sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)), are used to model periodic phenomena and to solve problems involving angles.

Each trigonometric function has specific properties, such as periodicity and symmetries, and particular values at key angles. For example:

• \(\sin(0) = 0\) and \(\cos(0) = 1\)
• \(\sin(\frac{\pi}{2}) = 1\) and \(\cos(\frac{\pi}{2}) = 0\)
• The functions repeat their values in regular intervals, known as their periods.

In the context of the exercise, \(\sec(x)\) is the reciprocal of \(\cos(x)\), which means that \(\sec(\frac{\pi}{2})\) (reciprocal of \(\cos(\frac{\pi}{2}) = 0\)) is undefined. However, because it is multiplied by 0 (the result of \(\ln(1)\)), the final product in step 4 is 0, leading to a well-defined limit.