Your doubts are genuine and perhaps they are a result of misinterpretation of definition of limit. The definition of limit is given in such a form that it can only be used to verify whether a number is a limit of some function at a point or not. It is not supposed to be used as a tool to find /evaluate limits. You can compare this situation with the fact that the definition of root of a polynomial hardly gives any practical method to find the root, but given a guess at the root one can use the definition to verify it.
Thus the standard approach in these problems is to guess the limit somehow and then use the definition to verify whether your guess is correct. How do I guess the limit then? Most problems give a specific number to be verified as a limit but this is not the case here and you need to guess the limit by using some calculation.
You may use your calculator and evaluate the function for values of $x$ near $2$ like for $x=2.1,2.01,2.001,1.9,1.99$ etc and find some pattern in the values of the function. If you are observant enough you will see that these values are all near the number $8$ and $8$ could be a good guess for the limit of the function under consideration. Now use the definition of limit to verify that $8$ is indeed the desired limit.
Also note that the above procedure of guessing and verifying a limit is practical only for very simple functions and giving such exercises for complicated functions is pointless. Using the definition of limit we can establish certain theorems (algebra of limits, squeeze, standard limits) which can be used to evaluate limits very efficiently. You should study the proofs of these theorems to understand the use of the definition of limit.