Problem 16 - APEX Calculus

Question

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 10} \cos (g(x)) $$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Determine the Value of g(x) Approaching 10

Given that \( \lim_{x \rightarrow 10} g(x) = \pi \) and \( g(10) = \pi \), we can conclude that as \( x \) approaches 10, \( g(x) \) approaches \( \pi \).

Step 2: Apply the Limit Function to \( \cos(g(x)) \)

Since \( g(x) \rightarrow \pi \) as \( x \rightarrow 10 \), and knowing that the cosine function is continuous, we can use the property of limits to evaluate: \( \lim_{x \rightarrow 10} \cos(g(x)) = \cos(\lim_{x \rightarrow 10} g(x)) = \cos(\pi) \).

Step 3: Calculate \( \cos(\pi) \)

The value of \( \cos(\pi) \) is \(-1\). Therefore, \( \lim_{x \rightarrow 10} \cos(g(x)) = -1 \).

Key Concepts

Continuity of functions

In calculus, the concept of continuity is intuitive and essential, especially when working with limits. A function is continuous at a point if there is no interruption during movement along the curve of the function at that point. Mathematically, for a function \( f(x) \), it is continuous at a point \( c \) if the following three conditions are met:\

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• \( f(c) \) is defined.
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• The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
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• The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).
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\These conditions imply that there are no jumps, breaks, or holes at the point \( c \). For instance, if \( \, g(x) \, \) has \( \lim_{x \to 10} g(x) = \pi \) and \( g(10) = \pi \), we understand that \( g(x) \) is continuous at \( x = 10 \).

Understanding this concept is crucial, as it allows us to use limit properties directly. In the step-by-step solution above, we assumed continuity to find the limit of \( \, \cos(g(x)) \, \) by evaluating it at \( \, g(10) = \pi \, \). Alternatively, for non-continuous points, we must evaluate each side's limit for the point.

Trigonometric functions

Trigonometric functions, such as cosine, sine, and tangent, are fundamental in calculus due to their periodic nature. These functions help model many real-world phenomena, like waves and oscillations. The continuity and differentiability of trigonometric functions are significant topics in calculus.

The cosine function \( \, \cos(x) \, \) is periodic with a period of \( \, 2\pi \, \), and it's continuous everywhere on the real number line. Meaning, at any given point \( x \), the function smoothly transitions without any jumps or breaks. This property is key when dealing with limits involving trigonometric functions.

In our exercise, the limit \( \lim_{x \to 10} \cos(g(x)) \) is calculated using the limit of \( \, g(x) \, \) itself. Since \( \cos(x) \) is continuous, we can write \( \lim_{x \to c} \cos(g(x)) = \cos(\lim_{x \to c} g(x)) \) whenever \( g(x) \) approaches a specific value. For instance, when \( x = 10 \), \( \lim_{x \to 10} g(x) = \pi \), leading to \( \lim_{x \to 10} \cos(g(x)) = \cos(\pi) = -1 \).

Working with trigonometric functions requires good understanding of their characteristics, as they include period, amplitude, and phase shifts, all crucial for interpreting limits effectively.

Limit properties

Limits are foundational in calculus, providing a method to evaluate how functions behave near certain points. Certain properties of limits make many otherwise complex calculations manageable. Key limit properties include:\

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• Sum/Product Rule: \( \lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \) and similarly for products.
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• Composition of Limits: If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \), then \( \lim_{x \to c} (f(g(x))) = f(M) \), provided \( f \) is continuous at \( M \).
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• Constant Multiplication: \( \lim_{x \to c} (k \, f(x)) = k \, \lim_{x \to c} f(x) \) for any constant \( k \).
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\By leveraging these properties, calculus harnesses the continuity of functions such as trigonometric functions. In our problem, once we know \( g(x) \) approaches \( \pi \), and knowing \( \cos(x) \) is continuous, \( \cos(g(x)) \)'s limit can be directly computed by applying these properties: \( \lim_{x \to 10} \cos(g(x)) = \cos(\lim_{x \to 10} g(x)) = \cos(\pi) \). Utilizing these limit properties enables us to transition smoothly from the given function \( g(x) \) to its cosine, simplifying our computations.