Question

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

Short answer

The limit \( \lim_{x \rightarrow 1} f(x) g(x) \) is 0.

Step-by-step solution

Understand the Limit Expression

The problem asks us to find \( \lim_{x \rightarrow 1} f(x) g(x) \), which means we need to evaluate the limit of the product of \( f(x) \) and \( g(x) \) as \( x \) approaches 1.

Identify Known Limits

From the given information, we know that \( \lim_{x \rightarrow 1} f(x) = 2 \) and \( \lim_{x \rightarrow 1} g(x) = 0 \). These are the limits of the functions \( f(x) \) and \( g(x) \) individually as \( x \) approaches 1.

Apply Limit Product Rule

The limit of the product \( f(x)g(x) \) as \( x \rightarrow 1 \) can be found by multiplying the limits of \( f(x) \) and \( g(x) \), provided both limits exist. Thus, \( \lim_{x \rightarrow 1} f(x)g(x) = \lim_{x \rightarrow 1} f(x) \cdot \lim_{x \rightarrow 1} g(x) = 2 \cdot 0 \).

Compute the Product

By computing \( 2 \cdot 0 \), we get the final result of the limit, which is 0.

Key concepts

Limit of a Function

In calculus, the limit of a function is a fundamental concept used to describe the behavior of a function as it approaches a particular input value. Specifically, the limit attempts to find what output the function is approaching as the input nears a specific point. This is expressed as \( \lim_{x \rightarrow a} f(x) \), which reads "the limit of \( f(x) \) as \( x \) approaches \( a \)."

Understanding limits helps to analyze functions at points where they might not be directly evaluated, such as points of discontinuity or locations where the function's expression becomes undefined. When we say the limit exists, it means that as we get arbitrarily close to \( a \), \( f(x) \) approaches a particular value \( L \).

It's important to note that for a limit \( \lim_{x \rightarrow a} f(x) \) to exist, the function must approach the same value from both the left and the right as \( x \) nears \( a \). If these limits are different, or if either does not exist, then the limit will not exist.

Product Rule for Limits

The product rule for limits is a property in calculus that simplifies the evaluation of limits for the product of two functions. If you have two functions, \( f(x) \) and \( g(x) \), and you know their limits individually as \( x \rightarrow a \), you can find the limit of the product \( f(x)g(x) \) by multiplying their limits. This is expressed as:

• If \( \lim_{x \rightarrow a} f(x) = L \) and \( \lim_{x \rightarrow a} g(x) = M \), then \( \lim_{x \rightarrow a} f(x)g(x) = L \cdot M \).

However, this rule is applicable only when both limits \( L \) and \( M \) exist. If one or both of these limits are infinite or do not exist, then we cannot use this rule directly.

In the original exercise, we applied the product rule where \( \lim_{x \rightarrow 1} f(x) = 2 \) and \( \lim_{x \rightarrow 1} g(x) = 0 \). By the product rule, \( \lim_{x \rightarrow 1} f(x)g(x) = 2 \cdot 0 = 0 \). It's a straightforward example of how two limits, when known, can provide the limit of their product.

Functions in Calculus

Functions in calculus are mathematical entities that map inputs to outputs according to specific rules. Functions are central to calculus, and they help us model real-world phenomena and solve complex problems.

In calculus, we deal with different kinds of functions, including polynomial, trigonometric, exponential, and others. Analyzing these functions involves understanding their limits, derivatives, and integrals which help describe their behavior under various transformations and over different intervals.

• Limit of a Function: As discussed, it's essential for understanding behavior near specific points.
• Continuity: A function is continuous at a point if its limit equals its value at that point.
• Differentiability: A function is differentiable if it has a defined derivative at a point.

Through functions, calculus helps to resolve complex problems such as calculating areas under curves, finding instantaneous rates of change, and modeling naturally occurring patterns. Understanding these concepts is vital for mastering calculus.