To find the limit of a function as \( x \) approaches a specific value, one effective method is direct substitution. This approach is particularly useful for many functions, including polynomials and basic root functions, where the limit at a point is equal to the function's value at that point. For instance, if you want to find the limit of a polynomial function like \( 6x^3 + 3x^2 - x + 5 \) as \( x \) approaches 2, you can simply substitute 2 into the function. This results in:

\[6(2^3) + 3(2^2) - 2 + 5 = 6(8) + 3(4) - 2 + 5 = 48 + 12 - 2 + 5 = 63\]

Thus, the limit as \( x \) approaches 2 is 63.

Similarly, for a root function such as \( \sqrt{7x^2 + 4x + 16} \) as \( x \) approaches 0, direct substitution yields:

\[\sqrt{7(0^2) + 4(0) + 16} = \sqrt{16} = 4\]

Therefore, the limit is 4.

When dealing with rational functions, you can also use direct substitution, provided that the denominator does not equal zero at the point of interest. For example, to find the limit of \( \frac{x^2 + 3x + 2}{x + 1} \) as \( x \) approaches 0, first check the denominator:

\[0 + 1 = 1 \quad (\text{not } 0)\]

Since the denominator is not zero, you can substitute directly:

\[\frac{0^2 + 3(0) + 2}{0 + 1} = \frac{2}{1} = 2\]

Thus, the limit as \( x \) approaches 0 is 2.

In summary, for many functions, especially polynomials and basic roots, the limit can be found by simply substituting the value of \( x \) into the function. However, caution is needed with rational functions to ensure the denominator does not equal zero at the point of substitution. This method of direct substitution simplifies the process of finding limits significantly.