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# Evaluate the limit, if it exists. $\displaystyle \lim_{x \to 3}\frac{\frac{1}{x}-\frac{1}{3}}{x - 3}$

## $-\frac{1}{9}$

Limits

Derivatives

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

##### Michael J.

Idaho State University

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### Video Transcript

So here we have a specific limit. We have the limit as X purchased three of one divided by x -1 3rd over X -3. So you can evaluate this by one of the simpler techniques we have, which would be multiplying the numerator and denominator by three X. Otherwise essentially we have an indeterminate form with zero in the numerator and zero in the denominator. So We can find that this is equivalent to the limit as X approaches three of 3 -1 Divided by three x times x -3. So one thing we can do here is take the uh bring a negative out in the numerator negative x minus three, Divided by three x times x -3. So we can cross our X -3. And this would be equivalent to the limit as X approaches three of negative one divided by three X. And we can now use direct substitution since we don't have our indeterminant form anymore. And this would be equivalent to -1 knife. Another way to value this should be by using a lot lower overall where we differentiate the numerator and differentiate the denominator. But in this case likely it would be more simple by just using our standard limit techniques. And this is our final answer

Johns Hopkins University

#### Topics

Limits

Derivatives

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

##### Michael J.

Idaho State University

Lectures

Join Bootcamp