Math Limits Explained with Easy Animations

Now let's explore the difference between one-sided and two-sided limits.On the left, we have a continuous function, f of x equals x squared.As we approach x equals 1 from both sides, the function values approach 1.The two-sided limit exists because both the left-hand and right-hand limits equal 1.On the right, we have a piecewise function that behaves differently on each side of x equals 1.As we approach from the left, the function approaches 2.But as we approach from the right, the function approaches 3.Since the left and right limits are different, the two-sided limit does not exist at x equals 1 for our piecewise function.Let's explore what happens when we look at limits as x approaches infinity.First, let's examine the function f of x equals one over x.Notice how this function has two important lines it never crosses: a horizontal asymptote at y equals zero, and a vertical asymptote at x equals zero.As x approaches positive infinity, one over x gets closer and closer to zero.The same happens as x approaches negative infinity - the function approaches zero from below.Now let's look at a different function: g of x equals x divided by x plus one.This function has a horizontal asymptote at y equals one, which it approaches as x goes to either positive or negative infinity.Let's review what we've learned about limits at infinity.Thanks for exploring limits at infinity with Spark.E!