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The conjugate of any binomial term a+b is just the same binomial, but with the sign flipped in the middle. So if the original binomial is a+b, then its conjugate is ab. If the original binomial is ab, then its conjugate is a+b.
It makes sense then that the conjugate method for solving limits involves multiplying both the numerator and denominator of a rational function by the conjugate of either its numerator or denominator.
If we're going to use this method, then we have to identify either the numerator or denominator of the original fraction as a binomial term. Then we'll find the conjugate of this binomial and, to avoid changing the value of the original function, make sure to multiply both the numerator and denominator by the same conjugate, so that in effect we're multiplying by 1.
This method is usually use when the binomial term includes a square root, because multiplying by the conjugate will cancel the square root and make it easier for us to solve for the limit.
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