Math Problem Statement
Solution
The given mathematical exercise consists of the function . Let's solve the three parts step by step:
a) Finding the domain of the function
The function is defined as:
We know that the function is not defined when the denominator is zero. So, we set to ensure the function is defined. The domain of the function is therefore all real numbers except zero:
b) Prove that for every ,
We simplify the function :
For , we can cancel the terms:
Thus, we have proven that for all .
c) Calculate
From part (b), we know that for . Therefore, we calculate the limit as :
This is a simple linear function, and as , the limit becomes:
Thus, .
Would you like more details on any of the parts or further clarifications?
Here are 5 related questions for practice:
- What happens to the function if you modify it to ?
- How would you approach finding the derivative of ?
- What would the limit be if the function were as ?
- What is the range of ?
- How would the domain change if ?
Tip: When simplifying rational expressions, always check for domain restrictions before canceling terms.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Functions
Limits
Formulas
f(x) = (x^2 - 2x) / x
lim (x → 0) f(x)
Theorems
Limit Theorem
Simplification of Rational Expressions
Suitable Grade Level
Grades 9-11