Welcome to an introduction to limits in calculus. In mathematics, 'lim' stands for 'limit'. A limit describes the value that a function approaches as the input approaches a specific point. The notation for a limit is written as 'limit of f(x) as x approaches a'. In this example, we have a function with a gap at x equals 2. As x gets closer and closer to 2 from either side, the function value approaches 1. Therefore, we say the limit of the function as x approaches 2 is 1, even though the actual function value at x equals 2 is 3.
Now let's explore how to evaluate limits. There are several methods we can use. The first method is direct substitution, which works when the function is continuous at the point of interest. For example, to find the limit of x squared as x approaches 2, we can simply substitute x equals 2 into the function. The second method is factoring, which is useful when direct substitution gives an indeterminate form like zero over zero. For instance, to find the limit of x squared minus 4 divided by x minus 2 as x approaches 2, we can factor the numerator as (x plus 2)(x minus 2) and cancel the common factor, giving us a limit of 4. Other methods include rationalization for dealing with radicals and L'Hôpital's rule for certain indeterminate forms.
Let's now explore one-sided limits. A one-sided limit approaches a point from only one direction. The left-hand limit, denoted as limit as x approaches a from the left, examines the function's behavior as we approach the point a strictly from values less than a. Similarly, the right-hand limit, denoted as limit as x approaches a from the right, considers the function's behavior as we approach a from values greater than a. A two-sided limit exists if and only if both the left and right limits exist and are equal. In this example, we have a piecewise function where the left limit at x equals 2 is 1, and the right limit is 3. Since these values are different, the two-sided limit doesn't exist at this point.
Now let's examine limits at infinity. These limits describe the behavior of a function as x grows without bound in either the positive or negative direction. The notation 'limit as x approaches infinity' examines what happens to the function as x becomes arbitrarily large in the positive direction. Similarly, 'limit as x approaches negative infinity' looks at the function's behavior as x becomes arbitrarily large in the negative direction. If either of these limits exists as a finite value, it represents a horizontal asymptote of the function. In this example, we have the rational function f(x) equals x squared plus 1 divided by x squared plus 2. As x approaches either positive or negative infinity, this function approaches 1, which is the horizontal asymptote. This happens because as x gets very large, the highest power terms dominate, and the ratio approaches 1.
Finally, let's explore the applications of limits in calculus. Limits are fundamental to calculus and have numerous applications. First, limits are used to define derivatives. The derivative of a function at a point is defined as the limit of the difference quotient as h approaches zero. This represents the instantaneous rate of change or the slope of the tangent line at that point. In our example, we're looking at the function f(x) equals x squared. At the point x equals 2, the derivative is 4, which is the slope of the tangent line. As we decrease h, the secant line approaches the tangent line. Limits are also used to define integrals as the limit of Riemann sums. Additionally, limits help us analyze function behavior near critical points and solve optimization problems by finding where derivatives equal zero.