In today's video, we'll discuss and illustrate several limit laws that enable you to combine simple or known limits to calculate or understand the behavior of new limits that can arise and appear at first sight to be difficult or complicated. You'll become experts in no time. The first main collection of laws tells us that limits respect arithmetic. Arithmetic operations include addition, subtraction, multiplication, and division. These laws allow you to bring limits inside so to speak any of these operations provided all the components as simply limits actually exist. The first law states that "The limit of a sum is the sum of the limits." Symbolically, the limit of f of x plus g of x, is the limit of f of x plus the limit of g of x. The second law says that "The limit of a difference is the difference of the limits." Expressed symbolically as before but with minus instead of plus. The third law says that "The limit of a product is the product of the limits." Expressed symbolically this way now using multiplication. It follows as a special case that the limit of a constant multiple is the constant multiple of the limits. Expressed symbolically like this, where k is a constant. We often say constants come out the front. The fourth law says that "The limit of a quotient is the quotient of the limits." Expressed symbolically like this. Provided that at no stage are you dividing by zero. This last one's very important because we're often needing to deal with fractions in calculus with a possibility of zero as that denominator can become a serious and delicate issue. Notice that in describing these laws I wasn't being explicit about what happens to x in order to be general to cover a range of possibilities. You can have x approaching some number a from either side, or approaching a number from the right, or from the left, the so-called one-sided limits. We can also have x heading off towards infinity, by which we mean getting arbitrarily large and positive, or minus infinity getting arbitrarily large and negative. There are a few basic building blocks and you can combine them with the limit laws to progress very rapidly in many cases. Firstly, the limit of x as x approaches a is clearly just a. This is confirmed by looking at the graph of y equals x which I've drawn but made a tiny hole at x equals a because we're interested in limiting behavior as x approaches a but doesn't necessarily actually reach a. As x approaches a on the horizontal axis, y equals x approaches a on the vertical axis. The limit of a constant c regardless of what x is doing is just the constant c. This is clear by looking at the graph of y equals c. Again I've made a hole at x equals a. As x approaches a on the horizontal axis, y is stuck always on c on the vertical axis. If x gets very large and positive or large and negative then its reciprocal 1 on x tends to zero, so that the limit of 1 on x in both cases is equal to zero. This is clear from the graph which is a hyperbola. As x moves further and further away from the origin in either direction, the value of y equals 1 on x gets closer and closer to zero. Let's look at some examples. Let's find the limit as x approaches 2 of 2x minus 1. By the second law it becomes a difference of the limits of 2x and of one as x approaches two, which becomes the limit of 2 multiplied by the limit of x all take away 1, which quickly becomes 3. You can see the effect of this visually. The graph of y equals 2x minus 1 is just a straight line and we place a hole in the line corresponding to two on the horizontal axis which x is approaching. However, the hole lines up exactly with the value three on the y axis which is the limiting value of 2x minus 1. In this case we can actually fill in the hole with x-coordinate two and y-coordinate being the limiting value as x approaches two. As another example, the limit of x squared as x approaches minus 2 is by limit a law the square of the limit of x which is minus 2 squared which is 4. This also makes sense graphically, the graph is a parabola. If we put a hole in the graph at minus two on the horizontal axis which x is approaching, then the hole lines up exactly with four on the y-axis, the limiting value of x squared. Again, it's as though we can fill in the hole perfectly with x-coordinate minus two and y-coordinate being the limiting value as x approaches minus two. The previous two examples are special cases of the general phenomenon. To find the limiting value of the polynomial, simply evaluate it at the input that x is approaching. This relates to a property called continuity which we discuss in the next video. As a more elaborate example, consider the limit of this cubic polynomial as x approaches one. Because the cubic is built up from x and constants by addition and multiplication, we can apply the limit laws to bring the limit inside the expression. This has precisely the effect of evaluating the polynomial at x equals one yielding in this case an answer of two. We don't need to know what the graph actually looks like. In general, if p of x is any polynomial then the limit of p of x as x approaches a is just p of a. The result of substituting a for x and evaluating the expression using simple arithmetic. Let's look at a more elaborate example involving a rational function that is a quotient of two polynomials and try to deduce some asymptotic behavior by using the limit laws. Here's a very complicated expression. We want to see what happens as x goes to infinity. The leading terms in the numerator and denominator involving x cubed have been circled as they will dominate the expression as x gets large. The remaining bits and pieces involving large powers of x start to look insignificant by comparison with the leading terms when x is large. In the limit we expect the answer to be unchanged by simply throwing away these lower order insignificant pieces to get the limit of 3x cubed divided by 2x cubed, but then this cancellation and we're just left with the limit of the constant three on two which is just three on two. So, for these heuristic reasons we expect the answer to be three on two. For a more thorough or careful analysis we can do some algebraic manipulation first and then apply limit laws. The trick is to first divide through everything in the numerator and denominator by the highest power of x that appears, x cubed. This doesn't change the overall value of the expression. But by limit laws we can bring the limit inside the expression and see what happens to each of the simple pieces that we have created. The constants three and two are unaffected, but each of the fractions with x, x squared, and x cubed in the denominator have zero as their limit as x gets arbitrarily large and everything quickly becomes this expression which evaluates to three on two, which is what we were expecting earlier for heuristic reasons. This tells us in fact, that the horizontal line y equals three on two will be an asymptote to the curve for this particular rational function. We now discuss another important law that enables one to get information by making comparisons between known and unknown limits, aptly called the squeeze or sandwich law. If we have f of x sandwiched in between g of x and h of x, by which we mean g of x is less than or equal to f of x, which is less than or equal to h of x, for all x near x equals a, and the limits of g of x and h of x exists and are equal to L as x approaches a, then the limit of f of x also exists and equals this common limit L. Imagine a curve for G sitting beneath a curve of H but they are pinched together at some point, and the curve for F, even though possibly fluctuating wildly is sandwiched in between, and the point where they all pinch together corresponds to x equals a and y equals L, then all three limits exist and equal L as x approaches a. For example, let's use this squeezing idea to investigate the important but difficult limit sine x on x as x goes to zero. This limit is crucial later for developing the derivative of the sine function. Let's do some exploration first to get a feel for what we might expect. If you type in sine one divided by one in your calculator, you get about 0.84. Sine 0.1 divided by 0.1 is about 0.998. Sine 0.001 divided by 0.001 is really close to one. As x gets closer to zero, the calculations suggest the sine x on x is getting close to one. So, I suspect the limiting value is one. Let's see if we can deduce this using general principles. Here's part of the unit circle with some angle in the first quadrant and we've extended the radius to meet the tangent line that touches the unit circle at the point one on the horizontal axis. Suppose that the angle, measured in radians, is x, so that the arc length along the unit circle subtended by the angle is x. The point where the line extending the radius meets the tangent is that height tan x above the horizontal axis. Moving across from the point on the unit circle to the vertical axis produces the value sine x. Now we join the point on the unit circle to the point one on the horizontal axis, producing the short secant. And now, extract this diagram involving two triangles and a wedge or sector from the unit circle sandwiched in between. If we shade in pink the smaller triangle, with base length one and height sine x, then we get an area of sine x on two. If we shade in green, the wedge or sector of the circle subtended by the angle x, then the area is the proportion, x over two pi, of the area of the whole unit circle which is pi, giving an area of x on two. If we shade in blue the area of the larger triangle that engulfs everything, with base length one and height tan x, then we get an area of tan x on two. If we compare the three areas, we get that the pink area is less than or equal to the green area, which is which is less than or equal to the blue area. Thus we get a sequence of inequalities, sine x on two is less than or equal to x on two which is less than or equal to tan x on two. So, take this inequality and multiply through by two and rewrite tan x as sine x over cos x, and divide everything through by sine x retaining the inequalities because sine x is positive in our diagram. Then reciprocate everything which reverses the inequalities, noting that everything is positive. Notice that sine x on x is squeezed or sandwiched in between one on the left and cos x on the right. These values on the left and right both tend towards the same limit one as x tends towards zero. This is obvious for one on the left which is constant, for cos x on the right, you can perhaps visualize in your mind's eye the curve for cos x. And imagine as x moves towards zero, the value of cos x tends towards one, producing a limiting value of one. But sine x on x is sandwiched in the middle. So, by the squeeze law has nowhere else to move but forced also to go to one. This demonstrates that the limit of sine x on x as x approaches zero is indeed one as we suspected. Note that this argument used angles in the first quadrant of the unit circle, so that x was approaching zero from above. But the same argument can be adjusted to give the same result also if the angle is negative, that is if x is approaching zero from below. Let's do another example that looks similar using tan x instead over x. We don't need to go through an argument using the squeeze law again, but can just utilize the result for sine x over x. First write tan x as sine x over cos x and then split the expression up into two factors; sine x on x multiplied by one on cos x. And then use the product and quotient limit laws to express this as the limit of the first factor sine x on x, by the limit of the second factor, which in turn is one over the limit of cos x. Both of these limits are one so the expression quickly evaluates to one. Let's do another example making things slightly harder still, asking for the limit of tan 2x over x as x tends to zero. Observe that we can rewrite the fraction by multiplying the top and the bottom by two and then bringing the constant two in the numerator out the front. So, this becomes two times the limit of tan x over 2x, which we can rewrite as tan y over y, by putting y equals 2x, and notice that y tends to zero as x tends to zero. But the limit of tan y on y as y goes to zero is just one, our previous result using y instead of x. So we get two times one which is two. We've made significant progress utilizing limit laws to deal with some quite difficult and tricky limits. We discussed and illustrated how limits respect arithmetic operations. So, the limits can be brought inside sums, differences, products and quotients, and how we can force certain limits using the Squeeze Limit law when some possibly wild expression is sandwiched in between two expressions that we can control, and already know in advance converge to the same limit. We apply the Squeeze Limit law to deduce that sine x over x tends to one as x tends to zero, and use this to evaluate some limits involving the tan function. Please read the notes and when you're ready please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
