Determine the function limit \[ \lim_{x\to 1}\,\frac{x^2-1}{x^2-2x+1}. \]

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Lesson Parent: 
Function Limits III: The Gory Details
Problem: 

Determine the function limit
\[
\lim_{x\to 1}\,\frac{x^2-1}{x^2-2x+1}.
\]

Answer: 

The limit does not exist.

SAMPLE SOLUTION

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Solution: 

Both $x^2-1$ and <Sign in to see all the formulas> are polynomials and we know that in general that polynomials are continuous so that

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and

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So this function limit looks more like

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and we can't immediately tell if it is zero or undefined.

But we do know that

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Now our definition of a function limit does not require $x$ to be at $1$. This means that $x-1$ will never be zero and we can divide the $x-1$'s into one another to get

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Now as $x$ approaches $1$ the numerator approaches $2$ and the denominator approaches zero. We have shown here that these types of limits don't exist. In fact, the limit is unbounded so we write

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