91Ó°ÊÓ

To estimate limits graphically, we look at the behavior of a function's graph as it approaches a specific point. In this exercise, we analyze the limit \( \lim _{h \rightarrow 0} \frac{\cos (3h)-1}{h} \). To do this, we plot the function \( f(h) = \frac{\cos(3h) - 1}{h} \).
To graph this function, utilize a graphing tool. Plot points for small values of \( h \), both positive and negative, to see what happens as \( h \) gets closer to zero. The main focus is to observe the trend at \( h = 0 \).
As you examine the graph, look for where the function seems to steady. This "flattening out" near \( h = 0 \) provides a visual clue to the limit's value. In this example, the graph indicates the function approaches a specific value as \( h \) nears zero.

The small-angle approximation helps in simplifying trigonometric functions when the angles involved are close to zero. For functions involving \( \cos(x) \), if \( x \) is small, a useful approximation is \( \cos(x) \approx 1 - \frac{x^2}{2} \).
In the problem, we have \( \cos(3h) \) where \( h \) approaches 0. Using the small-angle approximation, we can replace \( \cos(3h) \) with \( 1 - \frac{(3h)^2}{2} \). This approximation simplifies the original expression \( \frac{\cos(3h) - 1}{h} \) into a more manageable form.
Substituting the approximation results in \( \frac{\left(1 - \frac{(3h)^2}{2}\right) - 1}{h} \), simplifying to \(-\frac{9h}{2}\). This expression is much easier to handle and can be algebraically manipulated to determine the limit's behavior.

Algebraic verification involves confirming a limit's value without graphical aids. In this case, we use our small-angle approximation to simplify the original function \( \frac{\cos(3h) - 1}{h} \).
By substituting \( \cos(3h) \approx 1 - \frac{(3h)^2}{2} \), the function becomes \( -\frac{9h}{2} \). As \( h \) tends to 0, observe the behavior of \( -\frac{9h}{2} \). Here, as \( h \) approaches 0, \( -\frac{9h}{2} \) also approaches 0, confirming our graphically estimated limit.
This method gives a robust confirmation. It shows the power of using algebra to back up visual estimates, providing a better grasp on why this specific limit works. Combining graphical and algebraic approaches gives a complete understanding of the limit's solution.