Question

In Exercises 49 and 50 , determine the limit. Assume that $$\lim _ { x \rightarrow 4 } f ( x ) = 0$$ and $$\lim _ { x \rightarrow 4 } g ( x ) = 3$$ (a) $$\lim _ { x \rightarrow 4 } ( g ( x ) + 3 )$$ (b) $$\lim _ { x \rightarrow 4 } x f ( x )$$ (c) $$\lim _ { x \rightarrow 4 } g ^ { 2 } ( x ) \quad \quad$$ (d) $$\lim _ { x \rightarrow 4 } \frac { g ( x ) } { f ( x ) - 1 }$$

Short answer

The answers are (a) 6, (b) 0, (c) 9, and (d) doesn't exist.

Step-by-step solution

Compute the limit for part (a)

Compute the limit of \(g(x) + 3\) as \(x\) approaches 4. Use the fact that the limit of a sum is the sum of the limits: \[\lim _ { x \rightarrow 4 } ( g ( x ) + 3 ) = \lim _ { x \rightarrow 4 } g ( x ) + \lim _ { x \rightarrow 4 } 3\]Replacing the limits already given, we get\[= 3 + 3 = 6.\]

Compute the limit for part (b)

For part (b), we compute the limit of \(xf(x)\) as \(x\) approaches 4. Use the property for the limit of a product: \[\lim _ { x \rightarrow 4 } x f ( x ) = \lim _ { x \rightarrow 4 } x * \lim _ { x \rightarrow 4 } f ( x )\]Replacing the limits already given, we get\[= 4 * 0 = 0.\]

Compute the limit for part (c)

For part (c), we compute the limit of \(g^2(x)\) as \(x\) approaches 4. Remember, the square of a function \(g(x)\) is simply \(g(x) * g(x)\): \[\lim _ { x \rightarrow 4 } g ^ { 2 } ( x ) = \lim _ { x \rightarrow 4 } g (x) * \lim _ { x \rightarrow 4 } g (x)\]Replacing the limits already given, we get\[= 3 * 3 = 9.\]

Compute the limit for part (d)

For part (d), we compute the limit of \(\frac{g(x)}{f(x) - 1}\) as \(x\) approaches 4. Remember that if a limit in the denominator is equal to 0, then the limit doesn't exist:\[\lim _ { x \rightarrow 4 } \frac { g ( x ) } { f ( x ) - 1 } = \frac{\lim _ { x \rightarrow 4 } g ( x )}{ \lim _ { x \rightarrow 4 } (f ( x ) - 1)}\]Replacing the limits already given, we get\[= \frac{3}{0 - 1}\]As we are dividing by zero, this limit doesn't exist.

Key concepts

Limit of a Sum

Understanding the limit of a sum in calculus is crucial when assessing how functions behave as they approach a certain point. It's based on a simple, yet powerful premise: if you can find the limits of individual functions separately as they approach the same value, you can simply add those limits together to find the limit of the sum of those functions.

For example, consider the function \( g(x) + 3 \) as \( x \to 4 \). You would find the limit of \( g(x) \) and the limit of the constant 3 as \( x \to 4 \). Since the limit of a constant is just the constant itself, adding these limits gives us the limit of the sum. It showcases how you can break down complex problems into simpler parts to find the solution.

Limit of a Product

The limit of a product follows a set of principles that are similar to the limit of a sum. If you are trying to find the limit of the product of two functions as \( x \to a \), you simply multiply the individual limits of those functions. This rule applies when both limits exist and are finite.

For instance, if we take the product \( xf(x) \) as \( x \to 4 \), we find the limit of \( x \) and \( f(x) \) as \( x \to 4 \) separately. Afterward, we multiply these two limits. This rule is a cornerstone of limit operations and provides a formulaic approach to solving limits that involve products.

Limits and Continuity

Limits and continuity are intertwined concepts in calculus. A function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. The idea of continuity extends beyond just a single point and speaks to the behavior of functions over an interval.

If a function has limits that do not match its value at the point of approach or has limits that are non-existent, then we're dealing with discontinuities. In the context of the given exercises, it's essential to recognize when limits result in undefined expressions, signaling a discontinuity, such as dividing by zero.

Solving Limits Step by Step

The process of solving limits step by step is fundamental for students to master. It involves analyzing a limit problem and applying known limit laws and properties systematically. To do this effectively, it's helpful to familiarize oneself with limits of common functions, theorems, and specialized techniques like factoring, conjugate multiplication, or L'Hôpital's Rule when applicable.

In our exercise, we applied limit laws for the sum, product, and the quotient of functions. Remember to account for special cases, such as a limit leading to an indeterminate form or a zero denominator, which requires alternative methods or might indicate the limit doesn't exist, respectively.