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

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Lesson Parent:

Problem:

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

Answer:

The limit does not exist.

Solution:

and

So this function limit looks more like

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

But we do know that

$x$ 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

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