Given $\left\{\begin{array}{l}\frac{x^3-1}{x^2-1}, \text { for } x<1 \\ \frac{3}{x-1}, \text { for } x \geq 1\end{array}\right.$. What is $\lim _{x \rightarrow 1^{-}} f(x)$?
Answer (2)
The limit of the function as x approaches 1 from the left is 2 3 , after simplifying the expression x 2 − 1 x 3 − 1 . By factoring the numerator and denominator, we find the limit evaluates to this value. Therefore, the answer is 2 3 .
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Factor the numerator and denominator of the expression x 2 − 1 x 3 − 1 to get ( x − 1 ) ( x + 1 ) ( x − 1 ) ( x 2 + x + 1 ) .
Cancel the common factor ( x − 1 ) to simplify the expression to x + 1 x 2 + x + 1 .
Evaluate the limit as x approaches 1: lim x → 1 − x + 1 x 2 + x + 1 = 1 + 1 1 2 + 1 + 1 .
The limit is 2 3 .
Explanation
Problem Analysis We are given a piecewise function f ( x ) and asked to find the limit as x approaches 1 from the left. This means we need to consider the part of the function defined for x < 1 .
Function Definition For x < 1 , the function is defined as f ( x ) = f r a c x 3 − 1 x 2 − 1 . To find the limit as x approaches 1 from the left, we need to evaluate l i m x r i g h t a rro w 1 − f r a c x 3 − 1 x 2 − 1 .
Factoring We can factor the numerator and the denominator of the expression. The numerator x 3 − 1 can be factored as ( x − 1 ) ( x 2 + x + 1 ) , and the denominator x 2 − 1 can be factored as ( x − 1 ) ( x + 1 ) . So, we have f r a c x 3 − 1 x 2 − 1 = f r a c ( x − 1 ) ( x 2 + x + 1 ) ( x − 1 ) ( x + 1 )
Simplification For x n e q 1 , we can cancel the ( x − 1 ) terms in the numerator and denominator, which gives us f r a c x 2 + x + 1 x + 1
Limit Evaluation Now, we can evaluate the limit as x approaches 1 from the left: l i m x r i g h t a rro w 1 − f r a c x 2 + x + 1 x + 1 = f r a c ( 1 ) 2 + 1 + 1 1 + 1 = f r a c 3 2
Final Answer Therefore, the limit of the function f ( x ) as x approaches 1 from the left is f r a c 3 2 .
Examples
Imagine you are designing a ramp where the height is given by the function f ( x ) = x 2 − 1 x 3 − 1 for x < 1 . To ensure a smooth transition at x = 1 , you need to know the limit of the height function as x approaches 1 from the left. This calculation helps you to design the ramp so that it smoothly connects to another part of the structure, avoiding any sudden changes in height.
