Tangents and Limits
A tangent to a curve is a straight line that touches the curve at a single point but does not intersect it at that point. For example, in the figure to the right, the y-axis would not be considered a tangent line because it intersects the curve at the origin. A secant to a curve is a straight line that intersects the curve at two or more points.
In the figure to the right, the tangent line intersects the curve at a single point P but does not intersect the curve at P. The secant line intersects the curve at points P and Q.
The concept of limits begins with the tangent line problem. We want to find the equation of the tangent line to the curve at the point P. To find this equation, we will need the slope of the tangent line. But how can we find the slope when we only know one point on the line? The answer is to look at the slope of the secant line. It's slope can be determined quite easily since there are two known points P and Q. As you slide the point Q along the curve, towards the point P, the slope of the secant line will become closer to the slope of the tangent line. Eventually, the point Q will be so close to P, that the slopes of the tangent and secant lines will be approximately equal.
A limit of a function is written as :
We want to find the limit of f(x) as x approaches a. To do this, we try to make the values of f(x) close to the limit L, by taking x values that are close to, but not equal to, a. In short, f(x) approaches L as x approaches a.
Left and Right Hand Limits
The previous example shows that the value a can be approached from both the left and right sides. Each side has its own limit. For example, as x approaches a from the right side we have 
and as x approaches a from the left we have 
.
The graph to the right shows an example of a function with different right and left hand limits at the point x = 1. As x approaches 1 from the left side, the limit of f(x) approaches 1. As x approaches 1 from the right side however, the limit of f(x) approaches 4. In this case, the limit of f(x) as x approaches 1 does not exist, because the left and right hand limits do not approach the same value. This idea leads to the following theorem:
For the limit to exist, the left and right hand limits must approach the same value. In our example, as x approaches 3, the left and right hand limits both approach a value of 4. Since the left and right hand limits are the same, the limit of f(x) as x approaches 3 exists and is equal to 4. Even though the actual value of f(3) is equal to 2, the limit is equal to 4. This gives the following theorem:
Infinite Limits
If a function is defined on either side of a, but the limit as x approaches a is infinity or negative infinity, then the function has an infinite limit. The graph of the function will have a vertical asymptote at a. A curve y=f(x) will have a vertical asymptote at x = a if any of the following conditions hold:
Note: When dealing with rational functions, there will always be a vertical asymptote at values of x that make the denominator equal to 0. This is due to the fact that the function is undefined at points where the denominator is 0. However, you must still show that one of the conditions above holds true to prove that there is a vertical asymptote at that point.
