Evaluate the following limit. `lim_(x->2)⁡ (1/(x-2)-1/(x^2-4))`          I used the rule for evaluating the limit of a difference. I want to confirm what the answer will be as I get 0.

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Combine the fractions to get

`1/(x-2)-1/(x^2-4)=(x+2)/(x^2-4)-1/(x^2-4)=(x+1)/(x^2-4).`

As ` ``x->2` from above, the numerator approaches 3 and the denominator is always positive and approaches 0, so the ` `expression grows without bound in the positive direction. That is,

`lim_(x->2^+)(1/(x-2)-1/(x^2-4))=oo.`

Similarly, we can show that` `

`lim_(x->2^-)(1/(x-2)-1/(x^2-4))=-oo,`

so the limit doesn't exist, even...

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Combine the fractions to get

`1/(x-2)-1/(x^2-4)=(x+2)/(x^2-4)-1/(x^2-4)=(x+1)/(x^2-4).`

As ` ``x->2` from above, the numerator approaches 3 and the denominator is always positive and approaches 0, so the ` `expression grows without bound in the positive direction. That is,

`lim_(x->2^+)(1/(x-2)-1/(x^2-4))=oo.`

Similarly, we can show that` `

`lim_(x->2^-)(1/(x-2)-1/(x^2-4))=-oo,`

so the limit doesn't exist, even in the form of `+-oo.` Remember that the rule for evaluating the limit of the difference of two expressions is valid only if both limits exist. Here, neither `1/(x-2)` or `1/(x^2-4)` has a limit as `x->2` , so the rule isn't valid.

 

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