Math Problem Statement
Hvis f(x) = { e^{1/x^2}, x ≠ 0, a, x = 0 }, hva må a være for at f skal være kontinuerlig for x = 0?
Solution
This problem is asking for the value of such that the function is continuous at . The function is defined as:
\begin{cases} e^{\frac{1}{x^2}}, & x \neq 0 \\ a, & x = 0 \end{cases}$$ To ensure continuity at $$x = 0$$, the following condition must be met: $$\lim_{x \to 0} f(x) = f(0) = a$$ ### Step 1: Evaluate the limit as $$x \to 0$$ Consider: $$\lim_{x \to 0} e^{\frac{1}{x^2}}$$ As $$x \to 0$$, $$\frac{1}{x^2} \to \infty$$, so: $$\lim_{x \to 0} e^{\frac{1}{x^2}} = e^{\infty} = \infty$$ ### Step 2: Analyze Continuity Since the limit tends to infinity, there's no finite value of $$a$$ that can satisfy: $$\lim_{x \to 0} f(x) = a$$ ### Conclusion: The function cannot be made continuous at $$x = 0$$ for any finite value of $$a$$. Would you like me to expand on any specific part of this explanation? --- ### Related Questions: 1. What is the formal definition of continuity at a point? 2. How do we evaluate limits involving exponential functions with singularities? 3. Can $$e^{1/x^2}$$ be made continuous by modifying its definition near zero? 4. What happens to the continuity if the exponent was $$-\frac{1}{x^2}$$ instead? 5. How does the behavior of $$e^{\frac{1}{x^2}}$$ compare to polynomial functions near zero? ### Tip: When dealing with continuity at a point, always check both the left-hand and right-hand limits. If either diverges, continuity can't be established.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Exponential Functions
Formulas
f(x) = e^{1/x^2}
lim_{x -> 0} f(x) = a
Theorems
Continuity at a Point
Limit Theorem
Suitable Grade Level
Grades 11-12
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