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When faced with the task of evaluating limits, one of the first strategies to consider is the Direct Substitution Method. This approach is exactly as it sounds; you simply substitute the value to which the variable is approaching into the expression to find the limit. This method works wonderfully when the function is continuous at the point of interest—meaning there's no interruption in its graph at that point. For example, in the given exercise, we have to evaluate \( \lim _{x \rightarrow 2}\left(x^{2}+2 x-7\right) \). To use direct substitution, we replace every instance of \( x \) with 2, essentially evaluating \( 2^2 + 2\cdot2 - 7 \) right off the bat. This process hinges on the function behaving 'nicely' at the point where \( x \) approaches 2. If the function was not continuous, or had some kind of undefined behavior at \( x=2 \), alternative methods would have to be used.

Calculating expressions is a key skill that involves executing mathematical operations in the correct order. Once you have replaced your variable with the approaching value using the direct substitution method, the next step is to perform the necessary calculations. Remember to follow the order of operations—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). In our exercise example, we substitute 2 into the expression and get \( 2^2 + 2\cdot2 - 7 \). Calculating in the correct order, we first compute the exponent to get 4, then perform the multiplication to get 4 and finally combine these results with subtraction: \( 4 + 4 - 7 \). When calculating expressions, especially more complex ones, it's important to work step by step to avoid mistakes and reach the correct answer.

After calculating the necessary expressions, the last step in evaluating limits is to simplify the results. Simplifying involves combining like terms and reducing the expression to its most basic form. It may also include factoring, canceling out terms, or using algebraic identities where appropriate. In our exercise, once we have calculated \( 4 + 4 - 7 \), we simply add and subtract the numbers to find the final result. Simplifying \( 4 + 4 - 7 \) results in \( 1 \), which tells us that the limit of \( x^2 + 2x - 7 \) as \( x \) approaches 2 is indeed 1. Simplifying not only gives us the precise value of the limit but also ensures that our final answer is presented in the clearest and most concise form possible, which is essential for understanding and communication in mathematics.