Question:

Find the limit of the following function algebraically:

Answer:

To find the limit algebraically, we can first try direct substitution to see if it yields a finite value. Let's substitute $x = 3$ into the function:

Since the result is an indeterminate form, we can use algebraic manipulation to simplify the expression. We can factor the numerator using the difference of squares:

Now we can cancel out the common factor of $(x-3)$:

Now we can try direct substitution again:

Therefore, the limit of the function as $x$ approaches 3 is 6.