Estimating limit values from tables Consider the function f (x) = x-3 x2 + 2x - 15 x 2.9 Complete the table to find the limit as x - 3. 2.99 2.999 f(x) 3 3.001 3.01 3.1 Based on your findings, what are the values of each of these limits? lim f (x) = lim, f (x) = lim f (x ) = By evaluating certain types of functions at a particular value, we may not necessarily have a sufficient understanding of the function's behavior at a specific point. This is especially true for rational functions that contain discontinuities. This is why the idea of a limit is so important. Rather than just going directly to some x - value using direct substitution, we can approach a point from either side to get a sense of behavior in the surrounding area. Looking at this 1928 historical photo of the New Tyne Bridge we can see where the missing section will be constructed. The actual height when finished would be the limit. With limits we can discuss behavior at a point whether that point exists or not. With limits we look at the y-value the graph approaches not what the actual y-value at that point. When finding limits ask "What is happening to y as it approaches a certain number?" You are findin the y - value for which the function is approaching as x approaches c. Limit Existence Theorem. lim f(x) exists only if lim f(x) = lim f(x) = L Where L is a real number. x-ct x -C Verbally: The limit as x approaches c on f (x) will exist if and only if the limit as x approaches c left is equal to the limit as x approaches c from the right