Question

In the accompanying figure determine the following limits when \(\alpha\) approaches 0 : $$ \lim \frac{R Q}{M P} $$

Short answer

Answer: The limit of the ratio $\frac{RQ}{MP}$ as $\alpha$ approaches 0 is 0.

Step-by-step solution

Identifying the given values

Let's analyze the accompanying figure. We are given: 1. \(RQ\) as a length 2. \(MP\) as another length 3. \(\alpha\), an angle in the figure. Our goal is to determine the limit of the ratio \(\frac{RQ}{MP}\) as \(\alpha\) approaches 0.

Obtain a relation between RQ and MP, and angle α

From the figure, observe the two triangles \(QRM\) and \(PRM\). These are two right triangles having angle \(MPR\) as a common angle and angle \(QRM\) as \(\alpha\)(given in the question). Using the tangent ratio of angles in a right triangle, we get: \ For triangle \(QRM\), $$ \tan(\alpha) = \frac{RQ}{RM} $$ For triangle \(PRM\), $$ \tan(\angle MRP) = \frac{MP}{RM} $$

Express RQ and MP as a function of α

From the relations obtained in step 2, we can write \(RQ\) and \(MP\) in terms of angle α: $$ RQ = RM \times \tan(\alpha) $$ and $$ MP = RM \times \tan(\angle MRP) $$

Express the ratio as a function of α

Now, we can express the ratio (\(\frac{RQ}{MP}\)) as a function of \(\alpha\): $$ \frac{RQ}{MP} = \frac{RM \times \tan(\alpha)}{RM \times \tan(\angle MRP)} $$ Simplifying the equation by canceling out the \(RM\) terms, we get: $$ \frac{RQ}{MP} = \frac{\tan(\alpha)}{\tan(\angle MRP)} $$

Take the limit of the ratio as α approaches 0

Finally, we'll take the limit of the ratio as \(\alpha\) approaches 0: $$ \lim_{\alpha \to 0}\frac{\tan(\alpha)}{\tan(\angle MRP)} $$ As \(\alpha\) approaches 0, \(\tan(\alpha)\) will also approach 0, therefore: $$ \lim_{\alpha \to 0}\frac{\tan(\alpha)}{\tan(\angle MRP)} = 0 $$ So, the limit of the given ratio \(\frac{RQ}{MP}\) as \(\alpha\) approaches 0 is 0.

Key concepts

Tangent Ratio

The tangent ratio is a fundamental concept in right triangle trigonometry. It involves the relationship between the length of the sides of a right triangle. Specifically, the tangent of an angle in a right triangle is the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. This can be expressed in the formula:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In the context of our exercise, when we are given angle \(\alpha\), the tangent ratio can be applied to find the relationship between the side lengths of the right triangle involving \(\alpha\). The concept is used to express one side length in terms of the other and the angle, which is essential for solving problems related to trigonometry and limits.

Right Triangle Trigonometry

Right triangle trigonometry deals with the relationships between angles and sides of right triangles. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
For any acute angle in a right triangle:

• The sine of the angle is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of the angle is the ratio of the length of the adjacent side to the hypotenuse.
• The tangent, as previously mentioned, is the ratio of the opposite side to the adjacent side.

These ratios allow us to solve for unknown sides and angles in right triangles. In our exercise, the tangent ratio is specifically used to establish a relationship between the lengths \(RQ\) and \(MP\) as functions of the angle \(\alpha\) and the common side \(RM\).

Calculating Limits

Calculating limits is a core operation in calculus, mostly dealing with the behavior of a function as its input approaches a certain value. A limit answers the question: As the variable gets infinitely close to a certain value, what value does the function approach?
When calculating limits analytically, we frequently manipulate the function algebraically to reach a form that allows us to directly substitute the approaching value of the variable. However, if direct substitution leads to an undefined expression such as division by zero, we may have to use other techniques like factoring, rationalizing, or applying special limit laws to find the limit. In our exercise, we are interested in the limiting behavior of the tangent ratio as the angle \(\alpha\) approaches zero.

Approaching Zero Limits

The concept of limits as values approach zero is particularly significant because it helps us understand the behavior of functions around the origin. The limits involving trigonometric functions as the angle approaches zero are often intuitive; for example, since tangent is the ratio of sine to cosine, and as the angle approaches zero, sine approaches zero faster than cosine, the tangent of an angle will also approach zero.
In our case, we consider the limit of \(\tan(\alpha)\) as \(\alpha\) approaches zero. The limit is:\[ \lim_{\alpha \to 0} \tan(\alpha) = 0 \]

• If \(\tan(\alpha)\) approaches zero, then any ratio that has \(\tan(\alpha)\) in the numerator will also approach zero, provided the denominator is not concurrently approaching zero or does not equal zero.

This understanding aids in determining the limit of the ratio \(\frac{RQ}{MP}\) as \(\alpha\) approaches zero in the exercise.