x-2 51. lim x-1x-1

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the one-sided limits

### Problem 51
Evaluate the following limit as \( x \) approaches 1 from the left:
\[ \lim_{{x \to 1^-}} \left( \frac{x - 2}{x - 1} \right) \]

To solve the limit, consider the behavior of the function \(\frac{x - 2}{x - 1}\) as \( x \) approaches 1 from the left.

#### Solution:
1. **Substitute a value close to 1 from the left into the expression \(\frac{x - 2}{x - 1}\):**
    - For \( x < 1 \), \((x - 1)\) is a small negative number.
    - Hence, \(\frac{x - 2}{x - 1}\) evaluates the value.
    - As \( x \) gets closer to 1 from the left side, \((x - 2)\) approaches \(-1\) and \((x - 1)\) approaches a very small negative number.

2. **Analyze the fraction:**
    - \(\frac{x - 2}{x - 1} = \frac{-1}{\text{small negative number}}\)
    - Since dividing a negative number by another negative number results in a positive number, and the denominator is approaching zero.

The value of the expression becomes very large positively. 

Thus,
\[ \lim_{{x \to 1^-}} \left( \frac{x - 2}{x - 1} \right) = +\infty \]

Therefore, the limit of \(\frac{x - 2}{x - 1}\) as \( x \) approaches 1 from the left is \( +\infty \).
Transcribed Image Text:### Problem 51 Evaluate the following limit as \( x \) approaches 1 from the left: \[ \lim_{{x \to 1^-}} \left( \frac{x - 2}{x - 1} \right) \] To solve the limit, consider the behavior of the function \(\frac{x - 2}{x - 1}\) as \( x \) approaches 1 from the left. #### Solution: 1. **Substitute a value close to 1 from the left into the expression \(\frac{x - 2}{x - 1}\):** - For \( x < 1 \), \((x - 1)\) is a small negative number. - Hence, \(\frac{x - 2}{x - 1}\) evaluates the value. - As \( x \) gets closer to 1 from the left side, \((x - 2)\) approaches \(-1\) and \((x - 1)\) approaches a very small negative number. 2. **Analyze the fraction:** - \(\frac{x - 2}{x - 1} = \frac{-1}{\text{small negative number}}\) - Since dividing a negative number by another negative number results in a positive number, and the denominator is approaching zero. The value of the expression becomes very large positively. Thus, \[ \lim_{{x \to 1^-}} \left( \frac{x - 2}{x - 1} \right) = +\infty \] Therefore, the limit of \(\frac{x - 2}{x - 1}\) as \( x \) approaches 1 from the left is \( +\infty \).
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