How do you find #lim_(xto2) (x^36x2)/(x^34x)# using l'Hospital's Rule or otherwise?
The twosided limit doesn't exist...
graph{(x^3  6x  2)/(x^3  4x) [4.245, 9.8, 1.71, 5.313]}
Well, you can approach it with some manipulation.
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To find ( \lim_{x \to 2} \frac{x^3  6x  2}{x^3  4x} ), we can apply L'Hôpital's Rule.

Evaluate the limit directly: Substituting ( x = 2 ) into the expression, we get: [ \frac{2^3  6(2)  2}{2^3  4(2)} = \frac{8  12  2}{8  8} = \frac{6}{0} ]

As the denominator approaches zero, we can use L'Hôpital's Rule: Differentiate the numerator and denominator separately: [ \lim_{x \to 2} \frac{d}{dx}(x^3  6x  2) \Big/ \frac{d}{dx}(x^3  4x) ] [ = \lim_{x \to 2} \frac{3x^2  6}{3x^2  4} ]

Now, evaluate the limit again by substituting ( x = 2 ): [ \lim_{x \to 2} \frac{3(2)^2  6}{3(2)^2  4} = \frac{3(4)  6}{3(4)  4} = \frac{6}{8} = \frac{3}{4} ]
Therefore, ( \lim_{x \to 2} \frac{x^3  6x  2}{x^3  4x} = \frac{3}{4} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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