Continuity and limits

What Nour says

1. 1. Reminders

   Today, the whole chapter: "Continuity and limits", Year 4, Mathematics section.

   We start with reminders, then infinite branches, composed functions, order and limits, and end with images of intervals.

   Let's start with section one: reminders about continuity.

   A function f is continuous at a point a when its limit at a equals f of a.

   That is, the curve has no jump and no hole at that point.

   Here is a smooth curve, continuous everywhere we draw it.

   Now picture a curve with a jump at x equals two.

   The left piece stops, and the right piece starts somewhere else.

   This function is not continuous at x equals two, because the two sides disagree.

   A function continuous at every point of an interval is called continuous on that interval.

   Most familiar functions, like polynomials, are continuous on their whole domain.

   Now remember limits: a limit tells us where f goes as x approaches some value.

   It can also describe behavior as x goes to plus infinity or minus infinity.

   Let's work a full example to recall the main limit techniques.

   Consider f of x equals x squared minus one, divided by x minus one.

   We want the limit of f as x approaches one.

   Direct substitution gives zero over zero, an indeterminate form.

   So we factor the numerator, since it is a difference of squares.

   Now f of x simplifies to x plus one, for x different from one.

   Now we substitute directly, since this new form has no problem at one.

   So the limit of f as x approaches one equals two.

   Notice f itself is not defined at x equals one, it has a hole there.

   This gap between the limit and the actual value connects to extension by continuity (prolongement par continuité).

   That is, we can define a new function equal to f, plus the value two at one.

   This new function g is continuous everywhere, including at x equals one.

   We fill the hole, and the curve becomes complete and smooth.

   So the key idea: continuity means no jumps, no holes, limit equals function value.

   Limits let us study points where a function is not directly defined.

   And sometimes we can repair that point, using extension by continuity, to get a continuous function.

   Keep this example in mind, we will build on it all through this chapter.

   What is the limit of (x²-4)/(x-2) as x approaches 2?

2. 2. Infinite branches

   Now let's move to section two: infinite branches of a curve.

   An infinite branch happens when x or f of x goes to infinity.

   There are several shapes of infinite branches, and each has its own picture.

   First, a vertical asymptote: f of x goes to infinity as x approaches a fixed value.

   Here f of x equals one over x minus two, near x equals two.

   The dashed line x equals two is the vertical asymptote of this curve.

   Second, a horizontal asymptote: f of x approaches a fixed value l as x grows large.

   Here f of x equals two plus one over x, approaching two far away.

   The dashed line y equals two is the horizontal asymptote here.

   Third, an oblique asymptote: the curve gets close to a slanted line, not flat or vertical.

   Here f of x equals x plus one over x, close to the line y equals x.

   We check this by showing f of x minus x tends to zero at infinity.

   Let's fully solve a typical bac-style example now.

   Let f of x equal two x plus three plus one over x minus one.

   We study the branch of the curve as x goes to plus infinity.

   First we compute f of x minus the line two x plus three.

   As x goes to infinity, one over x minus one tends to zero.

   So the line y equals two x plus three is an oblique asymptote.

   The sign of one over x minus one tells us the curve's position.

   For large x, this term is positive, so the curve stays above the line.

   So, remember three types: vertical, horizontal, and oblique infinite branches.

   Each type comes from a different limit, at a point or at infinity.

   Finding the right limit tells you which asymptote the curve follows.

   What is the horizontal asymptote of f(x) = 3 + 1/(x-2) as x tends to infinity?

3. 3. Continuity and limit of a composite function

   Now section three: continuity and limit of a composed function.

   A composed function means applying one function, then another, to the same input.

   We write g of f, meaning first f acts on x, then g acts on the result.

   There is a rule for the limit of a composite function (limite d'une fonction composée).

   That is, if f tends to some value b, and g tends to a limit l at b, then g of f tends to l.

   Think of it as a relay race: f passes the baton to g at the point b.

   Here x approaches a, f of x approaches b, then g carries b toward l.

   If both f and g are continuous, then g composed with f is also continuous.

   That is, if f is continuous at a and g is continuous at f of a, then g of f is continuous at a.

   This means the composition of continuous functions never creates a jump or a hole.

   Here is a limit case: if g has a jump exactly at b, the composed limit can fail.

   That is why we always check continuity before applying this composed limit rule.

   Now let's fully solve a bac-style example using this method.

   Let f of x equal the square root of x squared plus one.

   We want the limit of f as x goes to plus infinity.

   We see f as a composition: first u of x equals x squared plus one.

   Then we apply the square root function to u of x.

   As x goes to plus infinity, u of x goes to plus infinity too.

   The square root function tends to plus infinity as its input grows without limit.

   By the composed limit rule, f of x also tends to plus infinity.

   We chain the two limits through the intermediate value, exactly as the rule says.

   Let's also check continuity for this same function f.

   u of x is a polynomial, so it is continuous everywhere.

   The square root is continuous on all nonnegative numbers, and u of x is always positive.

   So f, the composition, is continuous on the whole set of real numbers.

   This confirms our earlier calculation, and shows no hidden jumps anywhere.

   So remember: to find a composed limit, split the function into simple pieces.

   Find the limit of the inner piece first, then apply the outer function to that value.

   And to check continuity of a composition, check continuity of each piece separately.

   Combine both continuous pieces, and the whole composed function is continuous too.

   Given f(x) = (2x+1)², what is the limit of f(x) as x approaches 0?

4. 4. Limits and order

   Now section four: limits and order between functions.

   When one function stays smaller than another, their limits keep that same order.

   There is also a powerful squeeze result called the sandwich rule.

   That is, if f is squeezed between two functions with the same limit, f shares that limit.

   There is a related idea called the limit of a monotonic function (limite d'une fonction monotone), for increasing or decreasing functions.

   That is, an increasing function bounded above always has a finite limit at infinity.

   Picture two curves hugging a third curve tightly, forcing it toward the same value.

   Here g of x equals negative one over x, and h of x equals one over x.

   Both tend to zero as x grows, so anything squeezed between them also tends to zero.

   Here f of x equals sine of x over x, trapped between the two orange curves.

   Now let's solve a full bac-style example with this sandwich rule.

   Let f of x equal two plus cosine of x, all divided by x, for x bigger than zero.

   We find the limit of f as x goes to plus infinity.

   We know cosine of x always stays between negative one and one.

   Adding two to each part, we get one, three as bounds for two plus cosine.

   We divide everything by x, which is positive, so the inequalities stay the same way.

   Both one over x and three over x tend to zero at infinity.

   By the sandwich rule, f of x is squeezed to zero as well.

   This technique works perfectly when direct substitution or factoring fail to find a limit.

   So remember: order between functions carries over to order between their limits.

   The sandwich rule traps a function between two others sharing the same limit.

   A monotone bounded function always settles toward a finite limit, never escaping to infinity.

   If -x² ≤ f(x) ≤ x² for all x, what is the limit of f(x) as x approaches 0?

5. 5. Image of an interval by a continuous function

   Now section five: the image of an interval by a continuous function.

   When f is continuous on an interval, the set of all its values is also an interval.

   There is a key result here called the intermediate value theorem (théorème des valeurs intermédiaires).

   That is, if f is continuous on a closed interval, it takes every value between f of the endpoints.

   A special case is when f of a and f of b have opposite signs.

   This also needs a function continuous on a closed bounded interval, called the segment.

   That is, a closed bounded interval includes both its endpoints and does not stretch to infinity.

   On such a segment, a continuous function always reaches a minimum and a maximum value.

   Here is a continuous curve from a to b, crossing the x axis somewhere inside.

   At x equals zero, f is negative, and at x equals four, f is positive.

   Since f changes sign, the theorem guarantees a point c where f of c equals zero.

   Let's solve a full bac-style example to find this value c.

   Let f of x equal x cubed plus x minus one, on the interval zero, one.

   We show the equation f of x equals zero has a solution in this interval.

   First, f is a polynomial, so it is continuous on the segment zero, one.

   We compute f of zero: zero cubed plus zero minus one equals negative one.

   We compute f of one: one cubed plus one minus one equals one.

   f of zero times f of one is negative one times one, which is negative.

   By the intermediate value theorem, there exists c in zero, one with f of c equals zero.

   This proves the equation has at least one solution inside the open interval.

   We cannot find the exact value of c with this theorem alone, only its existence.

   We could narrow it down further by testing values like zero point five.

   So remember: continuity turns an interval into another interval, no gaps appear in the image.

   The intermediate value theorem guarantees solutions to equations, without needing to solve them explicitly.

   A sign change between two points always signals a root somewhere between them.

   This closed bounded condition matters, since open intervals do not guarantee a maximum or minimum.

   Does f(x) = x² - 2 have a root between x=1 and x=2?

6. 6. Image of an interval by a strictly monotonic function

   Now section six, the last one: image of an interval by a strictly monotone function.

   A strictly monotone function is always increasing, or always decreasing, on an interval.

   For a strictly increasing continuous function, the image of a, b is f of a, f of b.

   For a strictly decreasing function, the endpoints simply swap their order.

   This result combines with the intermediate value theorem to give a stronger existence and uniqueness tool.

   That is, under these conditions, there is exactly one solution, not just at least one.

   Here is a picture: a strictly increasing curve carries the interval a, b onto f of a, f of b.

   Here f of x equals x squared, strictly increasing when x stays positive.

   At x equals one, f equals one, and at x equals two, f equals four.

   So the image of the interval one, two is exactly the interval one, four.

   Let's solve a complete bac-style example using strict monotonicity.

   Let f of x equal x cubed plus x minus one, on the interval zero, one, again.

   We show the equation f of x equals zero has exactly one solution here.

   f is a sum of x cubed and x, both strictly increasing functions on this interval.

   So f itself is strictly increasing, and it is continuous as a polynomial.

   We already found f of zero equals negative one, and f of one equals one.

   Since f is continuous, strictly increasing, and changes sign, zero has exactly one preimage.

   So there exists a unique c in zero, one with f of c equals zero.

   This is stronger than before: now we know the solution is unique, not just existing.

   Strict monotonicity is the extra ingredient that turns existence into existence and uniqueness together.

   So remember: a strictly monotone continuous function maps an interval exactly onto another interval.

   Increasing functions keep the endpoint order, decreasing functions swap it.

   Adding strict monotonicity to the intermediate value theorem gives a unique solution, not only one that exists.

   This combined result is one of the most used tools in bac exercises on equations.

   Is f(x) = x³ + 2x + 1 strictly monotone on ℝ, and why?

   That completes the whole chapter on continuity and limits.

   We moved from reminders and infinite branches to composed functions, order, and images of intervals.

   Practice the two exercises above to make these methods fully yours.