# How do you find #lim 1/x^2-1/x# as #x->0#?

You can simplify the function in this way:

Now:

So:

graph{1/x^2-1/x [-10, 10, -5, 5]}

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To find the limit of 1/x^2 - 1/x as x approaches 0, we can simplify the expression first.

1/x^2 - 1/x can be rewritten as (x - x^2) / (x^2 * x).

Next, we can factor out an x from the numerator, giving us x(1 - x) / (x^3).

Now, we can cancel out one of the x terms in the numerator and denominator, resulting in (1 - x) / (x^2).

Finally, as x approaches 0, the expression (1 - x) / (x^2) approaches positive infinity.

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To find ( \lim_{x \to 0} \frac{1}{x^2} - \frac{1}{x} ), we'll first simplify the expression:

[ \lim_{x \to 0} \frac{1}{x^2} - \frac{1}{x} = \lim_{x \to 0} \left( \frac{1}{x^2} \right) - \lim_{x \to 0} \left( \frac{1}{x} \right) ]

Now, we evaluate each limit separately:

[ \lim_{x \to 0} \frac{1}{x^2} = +\infty ]

[ \lim_{x \to 0} \frac{1}{x} = +\infty ]

So, ( \lim_{x \to 0} \frac{1}{x^2} - \frac{1}{x} ) is an indeterminate form of ( +\infty - +\infty ), which means we cannot determine the limit directly. This suggests that further algebraic manipulation is needed to evaluate the limit.

One common technique is to combine the fractions under a common denominator:

[ \frac{1}{x^2} - \frac{1}{x} = \frac{x - x^2}{x^2} ]

Now, we'll factor out ( x ) from the numerator:

[ x - x^2 = x(1 - x) ]

So, the expression becomes:

[ \frac{x(1 - x)}{x^2} ]

[ = \frac{1 - x}{x} ]

Now, let's evaluate the limit as ( x ) approaches 0:

[ \lim_{x \to 0} \frac{1 - x}{x} = \frac{1 - 0}{0} = \frac{1}{0} = +\infty ]

Therefore, ( \lim_{x \to 0} \frac{1}{x^2} - \frac{1}{x} = +\infty ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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