Problem 26 - Essential Calculus Early Transcendentals

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{2+10^{x}}{3-10^{x}}$$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Identify Behavior as x Approaches Infinity

As we evaluate the limit \( \lim _{x \rightarrow \infty} \frac{2+10^{x}}{3-10^{x}} \), we notice that \( 10^x \) becomes extremely large as \( x \) approaches infinity. Consequently, the terms "2" and "3" become negligible compared to \( 10^x \).

Step 2: Simplify the Expression by Dividing by 10^x

To capture the behavior of the expression better, divide both the numerator and the denominator by \( 10^x \):\[ \frac{\frac{2}{10^x} + 1}{\frac{3}{10^x} - 1} \].As \( x \to \infty \), the fractions \( \frac{2}{10^x} \) and \( \frac{3}{10^x} \) approach 0.

Step 3: Evaluate Each Term

In the simplified expression, as \( x \to \infty \),\( \frac{2}{10^x} \to 0 \) and \( \frac{3}{10^x} \to 0 \).This transforms the expression to:\[ \frac{0 + 1}{0 - 1} = \frac{1}{-1} = -1. \]

Step 4: State the Limit

The limit of the function as \( x \to \infty \) is determined from the simplified expression. The calculated value is \( -1 \).

Key Concepts

Infinity Behavior

When tackling calculus problems involving limits, understanding infinity behavior is crucial. As the variable approaches infinity, we evaluate how different terms in an expression behave. In our given exercise, \( 10^x \) becomes extremely large when \( x \) approaches infinity, while constant terms like 2 and 3 do not change.

This vast difference in magnitude between \( 10^x \) and the constants renders the latter negligible when \( x \) goes to infinity. Focusing on this dominant term \( 10^x \), helps us in recognizing the leading behavior of the entire expression. Identifying which terms grow significantly or shrink to insignificance aids in simplifying complex expressions and understanding the overall trend as \( x \) heads towards infinity.

Simplifying Expressions

Simplifying expressions is a key step in evaluating limits. In the given expression \( \frac{2+10^{x}}{3-10^{x}} \), we can simplify it by dividing both the numerator and the denominator by the dominant term, \( 10^x \).

This leads to:

• Numerator: \( \frac{2}{10^x} + 1 \)
• Denominator: \( \frac{3}{10^x} - 1 \)

As \( x \) approaches infinity, the fractions \( \frac{2}{10^x} \) and \( \frac{3}{10^x} \) both approach 0.
The simplified expression becomes \( \frac{0+1}{0-1} \), effectively reducing the complexity of the original expression and paving the way for easy evaluation of its limit.

Evaluating Limits

Evaluating limits involves determining what value an expression approaches as the variable gets infinitely large or infinitely close to a specific point. In our problem, after simplifying, we arrive at the expression \( \frac{0+1}{0-1} \), which straightforwardly simplifies to \( \frac{1}{-1} \).

Thus, when \( x \to \infty \), the entire expression converges to -1. Evaluate each term independently in the simplified form to see their behavior. By isolating terms that go towards 0, we focus on terms that remain constant or dominate the result, ensuring accurate computation of the limit.

Calculus Step-by-Step Solutions

Step-by-step solutions in calculus offer a clear path to solving even the most complex limit problems. Following a structured method gives insight into solving similar problems in future.

• First, identify the dominant behavior as the variable approaches infinity.
• Simplify the expression appropriately to focus on meaningful terms.
• Evaluate these terms independently to arrive at a conclusive result.
• Summarize your findings to clearly state the limit of the function.

Working through such examples methodically enlightens students on the general approach in calculus, making seemingly difficult problems far more approachable. Embracing these techniques can empower students to independently tackle a wide array of problems involving limits.