Limits and asymptotic behaviour

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1. 1. Infinite limits at +∞ · Infinite limits at −∞

   Today, the whole chapter: "Limits and asymptotic behaviour", Year 3, Mathematics section.

   We will cover all ten parts of the chapter, from basic limits to asymptotes.

   Each part has a drawing, a worked example, and a short question for you to think about.

   Let's start with infinite limits, that is, limits where a function grows without bound.

   When x gets bigger and bigger, we ask what happens to f(x).

   If f(x) gets bigger and bigger too, with no limit, we say the limit is plus infinity.

   This means: as x tends to plus infinity, f(x) also tends to plus infinity.

   We write it with this arrow notation on the board.

   The same idea works going the other way, toward minus infinity.

   Here x becomes very negative, but f(x) still climbs up without bound.

   Let's draw this on a graph to see it clearly.

   Take the function f(x) equal to x squared.

   Look at the right branch: as x moves right, the curve shoots upward.

   So the limit of x squared, as x tends to plus infinity, is plus infinity.

   Look now at the left branch: as x moves left, the curve also shoots upward.

   So the limit of x squared, as x tends to minus infinity, is also plus infinity.

   This is why both branches of this curve rise forever, without any ceiling.

   There is no number that stops the curve from climbing higher.

   Think of it like walking on a road that never flattens, always going up.

   No matter how far you walk, the road keeps rising.

   Now let's look at a function where the limit is minus infinity instead.

   Take g(x) equal to minus x squared.

   This time, both branches go down, without any floor.

   So the limit of minus x squared, as x tends to plus infinity, is minus infinity.

   And the limit, as x tends to minus infinity, is minus infinity too.

   Now let's solve a full bac-style example, step by step.

   Find the limit of f(x) equal to x cubed minus 2x, as x tends to plus infinity.

   Step one: write f(x) by factoring out the strongest power, x cubed.

   Step two: as x tends to plus infinity, the fraction 2 over x squared tends to zero.

   So the term inside the brackets tends to one.

   Step three: x cubed itself tends to plus infinity.

   A quantity tending to plus infinity, times a quantity tending to one, still tends to plus infinity.

   So the limit of f(x), as x tends to plus infinity, is plus infinity.

   This factoring trick works for any polynomial: keep only the strongest power.

   The weaker terms always fade away compared to the strongest one.

   Remember: for even powers like x squared, both branches go the same way.

   For odd powers like x cubed, the two branches go opposite ways.

   This is the key difference between even and odd power functions at infinity.

   For f(x) = x³, what is the limit of f(x) as x tends to minus infinity?

2. 2. Finite limits at +∞ or −∞ · Horizontal asymptotes

   Now let's look at finite limits, that is, limits where f(x) settles near one fixed number.

   As x grows very large, f(x) does not explode, it gets closer and closer to a number L.

   The same can happen as x tends to minus infinity, approaching some number L too.

   Let's draw this with a simple function.

   Take f(x) equal to one over x, plus one.

   As x grows large, one over x shrinks toward zero.

   So f(x) gets closer and closer to one.

   This dashed line, y equal to one, is called a horizontal asymptote.

   That is, a flat line that the curve approaches but never actually touches.

   The curve hugs this line more and more closely, far to the right and far to the left.

   This happens because the gap between f(x) and one keeps shrinking toward zero.

   Think of it like a plane landing: it gets lower and lower but the ground is the line it nears.

   Near x equal to zero, the curve behaves very differently, but that is for later.

   Now let's solve a full bac-style example.

   Find the limit of f(x) equal to the fraction, 3x plus 1 over x minus 2, as x tends to plus infinity.

   Step one: divide top and bottom by x, the strongest power in the denominator.

   Step two: as x tends to plus infinity, one over x tends to zero.

   And two over x also tends to zero.

   Step three: the top tends to three, and the bottom tends to one.

   So the limit of f(x), as x tends to plus infinity, is 3.

   This tells us the curve of f has a horizontal asymptote, the line y equal to three.

   Dividing by the strongest power works for any rational function like this one.

   The same reasoning holds as x tends to minus infinity.

   In that case too, the limit is often the same finite number, three.

   A function can have one asymptote on the right and another on the left.

   The two horizontal lines do not need to match.

   Remember: a finite limit at infinity always means a horizontal asymptote on the graph.

   The curve approaches this line but it never needs to cross it, far from the origin.

   Sometimes the curve even crosses its asymptote near the middle, then approaches it later.

   What matters for the asymptote is only the behavior far away, not close to the center.

   What is the horizontal asymptote of f(x) = (2x − 5)/(x + 1) as x tends to +∞?

3. 3. Infinite limits at a real number · Vertical asymptotes

   Now let's study infinite limits at a real number, that is, when x approaches a fixed value but f(x) explodes.

   Here x does not go to infinity, instead x gets close to some number a.

   But f(x) itself grows without bound, toward plus or minus infinity.

   We often need to look from the left side and the right side of a separately.

   These two one-sided limits can even be different from each other.

   Let's draw this with a simple function.

   Take f(x) equal to one over x.

   Look at x close to zero, but on the right side, with positive values.

   As x shrinks toward zero from the right, one over x grows very large.

   So the limit, as x tends to zero from the right, is plus infinity.

   Now look at x close to zero, but on the left side, with negative values.

   As x shrinks toward zero from the left, one over x becomes very negative.

   So the limit, as x tends to zero from the left, is minus infinity.

   This dashed vertical line, x equal to zero, is called a vertical asymptote.

   That is, a vertical line the curve gets closer and closer to, but never crosses.

   This is why the curve rushes up on one side and down on the other side near zero.

   Think of it like a cliff: the curve climbs up one wall and drops down the other.

   Now let's solve a full bac-style example.

   Find the limits of f(x) equal to one over x minus 2, as x tends to 2.

   Step one: look at x minus 2 when x is slightly bigger than 2.

   Then x minus 2 is a small positive number.

   Step two: one divided by a small positive number is a very large positive number.

   Step three: now look at x slightly smaller than 2.

   Then x minus 2 is a small negative number.

   One divided by a small negative number is a very large negative number.

   So the curve of f has a vertical asymptote, the line x equal to two.

   This method works whenever the denominator of a fraction tends to zero.

   Always check the sign of the denominator, from the left and from the right.

   That sign tells you whether the limit is plus or minus infinity.

   Remember: a vertical asymptote appears exactly where the function is undefined, like dividing by zero.

   The one-sided limits can be equal, both infinite, or completely different in sign.

   For f(x) = 1/(x − 3), what is the limit as x tends to 3 from the left?

4. 4. Operations on limits

   Now let's see how to combine limits, using operations like sum, product, and quotient.

   If f and g each have a limit, we can often find the limit of f plus g, f times g, or f over g.

   Most of the time, you just add, multiply, or divide the two limits directly.

   But some combinations give no clear answer right away.

   These tricky cases are called indeterminate forms.

   In other words, these forms do not tell you the answer just by looking at them.

   You must transform the expression first, often by factoring, before you can conclude.

   Think of infinity minus infinity: both parts are huge, but their difference could be anything.

   This is why we saw the factoring trick in section one, to avoid this trap.

   Let's draw a quick picture to remember the four danger forms.

   Now let's solve a full bac-style example.

   Find the limit of f(x) equal to x squared minus x, as x tends to plus infinity.

   Step one: notice x squared tends to plus infinity, and x also tends to plus infinity.

   This gives the form infinity minus infinity, which is indeterminate.

   Step two: factor out x, the strongest power.

   Step three: as x tends to plus infinity, x tends to plus infinity.

   And x minus one also tends to plus infinity.

   A quantity tending to plus infinity, times a quantity tending to plus infinity, tends to plus infinity.

   So the limit of f(x), as x tends to plus infinity, is plus infinity.

   Factoring turned an indeterminate form into a simple product.

   This same strategy works for most polynomial and rational problems.

   Remember: the four danger forms are infinity minus infinity, zero times infinity, infinity over infinity, zero over zero.

   Whenever you see one of them, stop and transform the expression before concluding anything.

   Which indeterminate form appears when computing the limit of x² − x³ as x tends to +∞?

5. 5. Limits of a polynomial or rational function · Limits of √f

   Now let's focus on limits of a polynomial function, a rational function, and a function with a square root.

   For a polynomial, the limit at infinity is always the limit of its strongest term alone.

   For a rational function, the limit at infinity is the limit of the ratio of the two strongest terms.

   For a function with a square root, we treat the inside part first, then take the root of its limit.

   This works whenever the limit inside the root is zero or positive.

   Let's solve a full bac-style example for a rational function.

   Find the limit of f(x) equal to 2x squared minus 1, over x squared plus 3, as x tends to plus infinity.

   Step one: keep only the strongest terms, 2x squared on top and x squared on the bottom.

   Step two: simplify this ratio by cancelling x squared.

   So the limit of f(x), as x tends to plus infinity, is two.

   This means the curve of f has the horizontal asymptote y equal to two.

   Now let's solve a second example, with a square root this time.

   Take g(x) equal to the square root of x minus 2.

   We want the limit of g(x), as x tends to plus infinity.

   Step one: look at the inside part, x minus 2.

   As x tends to plus infinity, x minus 2 also tends to plus infinity.

   Step two: taking the square root of something tending to plus infinity still gives plus infinity.

   So the limit of g(x), as x tends to plus infinity, is plus infinity.

   Look at the curve: it keeps rising, but much more slowly than a straight line.

   This slow, ever-rising shape is a typical look for a square root function.

   For polynomials, remember only the highest power of x ever matters at infinity.

   For rational functions, only the highest powers top and bottom matter, the rest fades away.

   For square roots, always find the limit inside first, then apply the root.

   Be careful: a square root is only defined when the inside part is zero or positive.

   Always check this condition before writing any limit with a square root.

   These three rules cover almost every limit calculation you will meet at infinity.

   What is the limit of f(x) = (3x² + x)/(x² − 5) as x tends to +∞?

6. 6. Oblique asymptotes

   Let's finish with oblique asymptotes, that is, slanted lines a curve gets close to at infinity.

   A curve has an oblique asymptote when f(x) minus a line, a x plus b, tends to zero.

   In other words, the curve and the line get closer and closer, though the curve is not flat.

   This happens often for rational functions where the top has one more degree than the bottom.

   To find a and b, we usually divide the numerator by the denominator.

   Let's draw this with a picture first.

   Take f(x) equal to x squared plus 1, all over x.

   We can write f(x) as x, plus one over x.

   As x tends to plus or minus infinity, one over x tends to zero.

   So f(x) gets closer and closer to the line y equal to x.

   This slanted green line is the oblique asymptote of the curve.

   This is why the curve hugs the line far away, but curves off near the center.

   Close to zero, the curve behaves very differently because of the one over x part.

   Now let's solve a full bac-style example, step by step.

   Find the oblique asymptote of f(x) equal to 2x squared minus 3x plus 1, over x minus 1.

   Step one: divide the numerator by the denominator.

   This gives a quotient, 2x minus 1, and a remainder of zero.

   Step two: the remainder term tends to zero as x tends to plus or minus infinity.

   Step three: so f(x) minus the line, 2x minus 1, tends to zero.

   This means the line y equal to 2x minus 1 is the oblique asymptote.

   The method is always the same: divide, then check the remainder tends to zero.

   Sometimes, instead of a straight asymptote, the curve follows a parabola far away.

   We call this case a parabolic branch, that is, when the curve climbs like a parabola, not a line.

   This happens for example when f(x) minus x tends to infinity, not to zero.

   In that case there is no oblique asymptote, just this curved far-away behavior.

   Remember: an oblique asymptote exists only when f(x) minus the line tends exactly to zero.

   If that difference tends to infinity instead, you get a parabolic branch, not a line.

   Always divide polynomials carefully, since the quotient gives you a and b directly.

   This closes our tour of limits and asymptotes, from infinite limits to slanted lines.

   What is the oblique asymptote of f(x) = (x² + 2x)/x?

   That completes our tour of limits and asymptotic behaviour, from infinite limits to oblique lines.

   Keep practicing factoring, dividing by the strongest power, and checking indeterminate forms carefully.

   These skills will support everything you study next, especially derivatives and function studies.