Math Problem Statement
Solve in without integral
Examine the continuity of the function f(x) at x = 1, where f(x) is defined as a piecewise function: f(x) = ln(x)/(x-1) for x ≠ 1 and f(x) = 1 for x = 1.
Solution
To examine the continuity of the given function at , we need to evaluate the limit of as approaches and compare it with .
The function is defined as:
\frac{\ln(x)}{x - 1} & x \neq 1, \\ 1 & x = 1. \end{cases}$$ To check continuity at $$x = 1$$, we need to ensure: 1. $$f(1)$$ is defined. 2. $$\lim_{x \to 1} f(x)$$ exists. 3. $$\lim_{x \to 1} f(x) = f(1)$$. Let's go through each step. ### Step 1: Evaluate $$f(1)$$ From the definition, $$f(1) = 1$$. ### Step 2: Find $$\lim_{x \to 1} f(x)$$ To evaluate $$\lim_{x \to 1} \frac{\ln(x)}{x - 1}$$, we recognize this as a form suitable for L'Hôpital's Rule since both the numerator and denominator approach $$0$$ as $$x \to 1$$. Using L'Hôpital's Rule: $$\lim_{x \to 1} \frac{\ln(x)}{x - 1} = \lim_{x \to 1} \frac{\frac{d}{dx} \ln(x)}{\frac{d}{dx} (x - 1)} = \lim_{x \to 1} \frac{\frac{1}{x}}{1} = \lim_{x \to 1} \frac{1}{x} = 1.$$ ### Step 3: Compare $$\lim_{x \to 1} f(x)$$ with $$f(1)$$ Since $$\lim_{x \to 1} f(x) = 1$$ and $$f(1) = 1$$, we conclude that: $$\lim_{x \to 1} f(x) = f(1).$$ ### Conclusion The function $$f(x)$$ is continuous at $$x = 1$$. Would you like more details on the steps or further explanations on L'Hôpital's Rule? Here are some related questions to consider: 1. What is the purpose of L'Hôpital's Rule in calculus? 2. How can we evaluate limits without L'Hôpital's Rule? 3. What are other methods for checking continuity of piecewise functions? 4. Can the result change if we alter the piecewise definition at a different point? 5. How does the natural logarithm function behave near $$x = 1$$? **Tip:** When approaching limits with indeterminate forms like $$0/0$$, consider applying L'Hôpital's Rule or expanding terms using series expansion if applicable.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
L'Hôpital's Rule
Formulas
Limit definition for continuity: \( \lim_{x \to a} f(x) = f(a) \)
L'Hôpital's Rule: \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \) (when \( f(c) = g(c) = 0 \))
Theorems
L'Hôpital's Rule
Suitable Grade Level
Grades 11-12