Question

In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 0 } \frac { 3 \sin 4 x } { \sin 3 x }$$

Short answer

The limit of the function \( \frac { 3 \sin 4 x } { \sin 3 x }\) as \(x\) approaches \(0\) is \(4\).

Step-by-step solution

Plot the Function

First graphically analyze the function by plotting \( y = \frac { 3 \sin 4 x } { \sin 3 x }\). Make sure to include \(x = 0\) in your x-interval since that's where we want the limit.

Look for the Limit Graphically

Locate \(x = 0\) on your graph and estimate the y-value of the function at this point. This gives a graphical intuition for the limit as \(x\) approaches \(0\).

Substitute x = 0 into the function

We get \( \frac { 3 \sin 0 } { \sin 0 }\), which is an indeterminate form \( 0 / 0 \). We cannot determine the limit with this form.

Use L'Hopital's Rule

Since we have an indeterminate form, apply L'Hopital's rule, which states that if the limit of a function as \(x\) approaches a certain value results in an indeterminate form \(0/0\) or \(\infty/\infty\), then that limit is equal to the limit of the derivatives. We take the derivative of the numerator and denominator to get \( \frac { d } { dx } ( 3 \sin 4 x ) = 12 cos 4x \) and \( \frac { d } { dx } ( \sin 3 x ) = 3 cos 3x \). Then the limit becomes \( \lim_{x \rightarrow 0} \frac{12 \cos 4x}{3 \cos 3x} \)

Substitute x = 0 in the new function

Now we have \( \frac{12 \cos 0}{3 \cos 0} = \frac{12}{3} = 4 \), which is the final limit.

Key concepts

Limits and Continuity

Understanding the concept of limits is a cornerstone of calculus. Limits help us define the behavior of functions as they approach specific points. In the given exercise, we were asked to determine \(\lim _ { x \rightarrow 0 } \frac { 3 \sin 4 x } { \sin 3 x }\). Continuity, on the other hand, tells us about the function's behavior around the point of interest: if a function is continuous at a point, then the limit as the function approaches that point will equal the function's value at that point. However, in cases where the function shows unexpected behavior, such as a break or a hole, the function is not continuous, and we need to use limits to understand its behavior. Even if the function is not continuous at a point, the limit can still exist, providing insights into the function's tendency near that point.

Understanding both concepts works hand in hand, as continuity implies that the limit exists, and finding limits often helps in determining continuity.

L'Hopital's Rule

When addressing challenging limits that result in indeterminate forms like \(0/0\) or \(\infty/\infty\), L'Hopital's Rule comes to the rescue. It is a technique that relies on derivatives to find limits. In our exercise, after graphing the function and substituting \(x = 0\), we encountered the indeterminate form \(0/0\). L'Hopital's Rule states that if the function's limit results in an indeterminate form, then the limit of the function is the same as the limit of the ratio of their derivatives, provided the second derivative is not zero or does not become an indeterminate form.

To apply L'Hopital's Rule, you differentiate the numerator and the denominator separately, then take the limit of that new fraction. If this new limit is determinate or infinity, it is the limit of the original expression. This rule dramatically simplifies the evaluation of what seemed like an intractable limit and gives us a clear pathway to the solution.

Graphical Limit Estimation

Graphical estimation is a valuable initial step in understanding limits. By plotting the function, as done in Step 1 of our exercise, you can visually assess the behavior of the function as it approaches a particular point. Observing the graph around \(x = 0\) can give an intuition about where the function's value is heading. In this case, one looks for the \(y\)-value that the function appears to be reaching as \(x\) approaches \(0\) from both sides.

While graphical estimation is useful, it is not precise and cannot always be solely relied upon. For example, subtle function behaviors near the point, such as rapidly oscillating values, are difficult to detect by sight. This is why after estimating graphically, it is always a good practice to confirm the limit algebraically or with other numerical methods.

Indeterminate Forms

Indeterminate forms are expressions that do not lead to a clear, definite limit. The most common indeterminate forms are \(0/0\), \(\infty/\infty\), \(0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), and \(1^\infty\). In our exercise, when we substitute \(x = 0\) into \(\frac { 3 \sin 4 x } { \sin 3 x }\), we encounter the indeterminate form \(0/0\). It's indeterminate because both the numerator and the denominator approach zero, and the behavior of the function as \(x\) approaches \(0\) isn't immediately clear.

Indeterminate forms require more sophisticated math tools to solve, such as L'Hopital's Rule, algebraic manipulation, or other techniques in calculus. Recognizing an indeterminate form is the first step to solving these tricky limits, and understanding the nature of these forms is essential in advancing through calculus.