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Chapter 2: Problem 14

Find the limits. \begin{equation}\lim _{x \rightarrow 1^{-}}\left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)\end{equation}

Short Answer

Expert verified
The limit is 1.

Step by step solution

01

Simplify the Expression

Consider the expression \(\left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)\). To simplify it, combine the fractions into a single fraction: \(\frac{(x+6)(3-x)}{x(x+1)\cdot 7}\).
02

Evaluate the Limits of Individual Parts

Evaluate the limits of each part of the simplified fraction as \(x\) approaches 1 from the negative side: \(\lim_{x \to 1^{-}} \frac{x+6}{7x(x+1)}, \lim_{x \to 1^{-}} (3-x)\).
03

Calculate Each Limit Separately

First calculate \(\lim_{x \to 1^{-}} \frac{x+6}{7x(x+1)}\). Substitute in the function: \(x=1\) results in \(\frac{7}{14} = \frac{1}{2}\). For \(\lim_{x \to 1^{-}} (3-x)\) when \(x = 1\), it results in \(2\).
04

Combine the Results of Limits

The entire limit expression becomes \(\lim_{x \to 1^{-}} \left(\frac{1}{2}\right)\left(2\right) = \frac{1}{2} \times 2 = 1\).
05

Conclusion on Limit

Therefore, \(\lim_{x \to 1^{-}} \left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)\) simplifies to \(1\) after evaluating the limits of its parts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding One-Sided Limits
In calculus, one-sided limits describe the value that a function approaches as the variable approaches a specific point from one side only. This means either from the left or the right.
For example, in our exercise, the notation \(x \to 1^{-}\) signifies that we are considering the limit as \(x\) approaches 1 from the left side only.
  • One-sided limits help us examine the behavior of functions that may not be continuous or have different rules on either side of a point.
  • They are crucial for understanding discontinuities and defining derivatives.
To evaluate these, we substitute values that closely approach the target point from the specified direction, in this case from left to right for values less than 1.
Simplifying Fractions in Limits
Fraction simplification is a vital skill in evaluating limits, especially for complex expressions like the one given.
By combining multiple fractions into a single fraction, we make it easier to substitute values into the expression.
  • The product of fractions involves multiplying numerators and denominators separately.
  • Our original expression is simplified to \(\frac{(x+6)(3-x)}{x(x+1)\cdot 7}\), where all terms are in one fraction.
This makes calculating the limit straightforward, giving us clear terms to work with as we approach \(x = 1^{-}\). Simplifying fractions often reveals cancellation opportunities, simplifying the problem even further.
Evaluating Limits Step by Step
Once simplified, a key step in solving limit problems is systematically evaluating each part. One needs to handle fractions and numbers separately before combining them.
  • Look at specific parts: evaluate \(\lim_{x \to 1^{-}} \frac{x+6}{7x(x+1)}\) and \(\lim_{x \to 1^{-}} (3-x)\).
  • Substitute \(x = 1^{-}\) into each part and simplify wherever possible. \(\lim_{x \to 1^{-}} \frac{x+6}{7x(x+1)} \,=\, \frac{1}{2}\) and \(\lim_{x \to 1^{-}} (3-x) \, = \, 2\).
Combining these evaluated limits will give you the overall limit. Calculating piece by piece makes it easier to troubleshoot mistakes or verify calculations at each stage.
Approaching from the Left
When we approach a limit from the left, it means taking values that are slightly less than the point of interest. This focuses on behavior just before reaching a specific number, like 1 in our problem.
  • Values are considered such that they incrementally increase towards but do not exceed the specified point.
  • This helps avoid undefined expressions from occurring in the calculation, like division by zero.
Recognizing behavior as approaching from either the left or the right helps determine continuity and the precise limit value. In our exercise, focusing on \(x \,\to\, 1^{-}\) identified the limit as 1, demonstrating the predictability of the function from that side.

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