Despite using technology all of the time, I am a late-adopter when it comes to teaching upper school math. I was convinced that an analytical approach to problem-solving was best, since it requires critical reasoning skills. What’s an analytical approach? It’s the kind you probably learned in school as well: for the equation x^2 = 3x+4, you move everything to the left-hand side, factor, and solve.
And this is great for strong math students, but some people think differently. For limits on rational expressions of x as x approaches infinity, the best way to convey what happens may be to just use really, really big values of x and plug the expression into a calculator. This type of approach is called numerical. For the example above, you can show how the x^2 value gets smaller and then larger again, and how 3x+4 just gets larger.
A graphical solution would involve graphing the left-hand and right-hand expressions and seeing where they cross. There are downsides, of course; you need to know that your graphing window includes all of the solutions you’re looking for, and while that point looks like (4,16), is it exactly (4,16)?
But while I used to look at the graphing calculators as a crutch to critical thinking skills, I now see it as a useful tool in allowing the student to use numerical and graphical methods to solve problems. And I feel this is beneficial because each method can strengthen the comprehension of the other methods. For a student who thinks visually, the use of the graphing calculator may aid in his understanding of why the analytical method works, and so on. So while I’m a little late in incorporating the graphing calculator into the precalculus curriculum, we’re doing it this year.