Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow 1} \sqrt{2 x}=\sqrt{2} $$

Step-by-Step Solution

Answer

The limit of \( \sqrt{2x} \) as \( x \) approaches 1 is \( \sqrt{2} \).

Step 1: Understand the Limit Expression

The problem requires us to evaluate the limit of the function \( \sqrt{2x} \) as \( x \) approaches 1. This means we need to find the value that \( \sqrt{2x} \) gets closer to as \( x \) gets closer to 1.

Step 2: Substitute the Limit Value into the Function

To solve this, substitute \( x = 1 \) into the expression \( \sqrt{2x} \). This gives us \( \sqrt{2(1)} \).

Step 3: Simplify the Expression

Calculate \( \sqrt{2(1)} \), which simplifies to \( \sqrt{2} \). Thus, \( \sqrt{2x} \) approaches \( \sqrt{2} \) as \( x \) approaches 1.

Key Concepts

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that helps us find the value a function approaches as the input (or variable) approaches a certain point. In our example, we need to evaluate the limit \( \lim_{x \rightarrow 1} \sqrt{2x} \). Evaluating a limit involves determining the behavior of a function as the variable gets closer to a specific value.
This requires understanding how the function behaves near the point rather than finding its exact value.

By evaluating the limit, we check the value that the function is approaching, which is crucial for analyzing functions that may not be easily simplified at points.

• This technique helps in finding derivatives and integrals, which are essential tools in calculus.
• It helps in understanding continuity and stability of functions.

Mainly, limit evaluation assures us about trends in the behaviors of functions under study.

Square Root Functions

Square root functions, such as \( \sqrt{2x} \), are a special type of radical function that often appear in calculus problems involving limits. They are characterized by their distinct curve shape and tendency to be defined only for non-negative input. These functions output the square root of their input value, \( x \). In the problem we have, \( \sqrt{2x} \) is our main focus.

• Square root functions typically have a domain of non-negative numbers because a negative square root does not yield a real number.
• They are continuous where they are defined, which aids in evaluating limits.

Their continuous nature often simplifies limit problems, as substitution of the limit point into the function immediately gives us the limit value, as seen in our exercise. Here, the evaluation was straightforward due to the function's stability around the value being approached.

Substitution in Limits

Substitution is a straightforward method used for limit evaluation when the function is continuous at the point being considered. In our exercise, substituting the approaching value directly into the function \( \sqrt{2x} \) is the key step. This technique works well for functions without undefined forms, like division by zero.
When you substitute \( x = 1 \) into the expression, you calculate \( \sqrt{2(1)} \), simplifying to \( \sqrt{2} \). This confirms that as \( x \rightarrow 1 \), \( \sqrt{2x} \rightarrow \sqrt{2} \).

The steps are straightforward:

• If the function is continuous at the value, directly substitute the number for the variable.
• Simplify the resulting expression to find the limit.
• If direct substitution leads to an indeterminate form, consider other techniques, such as factoring or rationalization.

Substitution is often the simplest and fastest way to evaluate limits when applicable.