![]() However, there is another type of asymptote (Horizontal asymptote) and if that is the case, them x= a is your vertical asymptote. In other words, you have a vertical asymptote if Meanwhile, the y value continues to increase. This is a vertical asymptote. The value x is approaching is the asymptote because the function goes closer and closer to that x value. By now, we have already covered one of the asymptotes you need to know for this course. For example, although the function below is piecewise, it is continuous.Ĭonnecting Infinite Limits and Vertical Asymptotes or Horizontal AsymptotesĪn asymptote is a line that approaches a given curve but does not meet it at any distance. If they are equal, then it would be continuous. These are very interesting because sometimes they can be continuous, but at other times, they cant be.Ī piecewise function cant be continuous if the values of the functions on both sides of a boundary (defined by the piecewise function) are not equal. On your exam, you may come across a piecewise function. This works for functions like polynomial, logarithmic, and exponential. (For instance: f(x)=x^2 for ), then the function will be continuous on that interval. If a function is defined only for a certain interval. We know a function is continuous at a certain point if the function continues and crosses that point without any break or interruption.įor instance, the function above is continuous at x=1 but not x=2. But just looking at the graph at f(x) will not give you the answer because there is a discontinuity there. ⚡️ Watch- AP Calculus AB/BC: Continuity, Part IĪs you can see, the double-sided limit here does exist because the function approaches the same value from both sides at x. If the limit of a certain function is too hard to solve for, you can solve for the limit of similar, easier functions to help you.ĭefining Continuity at a Point of Interval ⚡ Read- AP Calculus AB/BC: Squeeze TheoremĪnother way to solve for a limit is using the Squeeze Theorem. Then plug in the x value the limit approaches as x and that is your answer.ĭetermining Limits Using the Squeeze Theorem If you get 0/0, that means that you need to simplify the function before plugging in the x-value. When you have a double-handed limit to solve for, first try and plug in the value it is approaching as x. Not only can you find them on a graph or table, but you can find them algebraically! There are multiple ways we can solve for limits. For a left-handed limit, go from x values from negative infinity to 3 and for the right hand, look at the y values that correspond to the y values coming from 3 to positive infinity.ĭetermining Limits Using Algebraic Manipulation and Selecting Procedures for Determining Limits This can also be used to find single-handed limits. Since as x approaches 3, the y value is approaching 0.25, it is clear that as x approaches 3, the limit of the function on the table is 0.25. The function to the left should not influence your answer at all. If it has a positive sign, you only follow the function from your right-hand side and see what y value it approaches as it approaches a certain x value from the right side. Limits that have a positive or negative sign at the top of the x value it is approaching are called single-handed limits. Limits can be solved algebraically - simplify the function as much as possible, and when you are done, plug in the x value and you will get your answer! ![]() ![]() If you get an actual number, then that's your answer! ![]() Limits can be solved for on a graph- follow the function to as close to the x value it approaches as possible and then look at what value you have reachedĪlso, limits can also be solved for by plugging in the x value: Plug the x value into the function and if you get 0/0 or indeterminate, then solve for it algebraically. This means that as x approaches a, on the graph of f(x), the value on the y-axis that the function approaches is L ![]()
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