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The limits of accuracy for the number $0.789$ when rounded to two decimal places are $0.7845$ and $0.7935$.

Rounding a number to two decimal places means approximating it to the nearest hundredth. For the number $0.789$, the relevant digits in the two decimal places are $7$ and $8$. To establish the limits of accuracy, we need to identify the range of values that would round to $0.79$ when rounded to two decimal places.

To determine the lower limit, we must find the smallest number that would still round up to $0.79$. This value is $0.7845$, as any number less than $0.7845$ would round down to $0.78$. Conversely, the upper limit is the largest number that would round down to $0.79$. This value is $0.7935$, since any number greater than $0.7935$ would round up to $0.80$.

Thus, any number within the range from $0.7845$ to $0.7935$, including $0.7845$ but excluding $0.7935$, will round to $0.79$ when rounded to two decimal places. This range is referred to as the limits of accuracy for the rounded number. Understanding these limits is essential for estimating and maintaining precision in calculations, a vital skill in GCSE Maths.