Limits and continuity

Review functions and closeness

A function assigns exactly one output to every permitted input. The domain is the set of those inputs. In f(x)=1/(x−3), x=3 is excluded: division by zero does not define a real number. A graph represents the pairs (x,f(x)), not the values we would like to exist. Before calculating a limit, write the domain and locate the point being approached.

The symbol x→a means we choose inputs arbitrarily close to a, usually different from a. It does not necessarily describe an object moving through time. We can approach 3 through 2,9; 2,99; 2,999 or through 3,1; 3,01; 3,001. These two lists illustrate different sides, but a finite list alone never proves the behaviour of all nearby inputs. Interpreting a limit requires controlling the entire neighbourhood within the domain.

What it means for an output to approach a value

Writing lim_x→a f(x)=L means we can make f(x) as close to L as we wish by sufficiently restricting the distance from x to a, with x≠a. The desired precision in outputs determines the necessary closeness in inputs. For f(x)=2x+1 near a=3, we have |f(x)−7|=2|x−3|. For an output error below 0,02, an input error below 0,01 is enough.

This relationship shows the logic behind the rigorous definition: first choose a positive tolerance ε for outputs; then seek a positive tolerance δ for inputs. If 0<|x−a|<δ implies |f(x)−L|<ε, the limit is controlled. You do not need to memorise letters to begin, but to understand the order: guarantee any desired precision, rather than merely present apparently close values.

Both sides must agree

The left-hand limit uses inputs smaller than a; the right-hand limit uses larger inputs. When domain inputs exist arbitrarily close on both sides, the finite two-sided limit exists precisely when both one-sided limits exist and are equal. If a function equals −2 to the left of 0 and 4 to the right, there is no single number describing both approaches. Assigning f(0)=1, or any other value, does not resolve that jump.

At a domain endpoint, continuity can be analysed only from the permitted side. For example, √x is continuous at 0 on the domain [0,+∞), using a right-hand approach. We should not invent negative inputs to demand a real two-sided limit that the domain itself does not allow. In your answer, state whether you are using a two-sided limit or an approach relative to the domain.

Algebra that reveals the behaviour

For polynomials, substituting the input gives the limit, because sums and products preserve finite limits. For a quotient, direct substitution also works if the denominator's limit is nonzero. When 0/0 appears, you have neither obtained a value nor demonstrated nonexistence: you have found an indeterminate form requiring further analysis.

In (x²−25)/(x−5), factor the numerator: (x−5)(x+5). For x≠5, the expression equals x+5; therefore the limit near 5 is 10. Simplification does not make the original expression defined at 5. With roots, multiplying by the conjugate may reveal the same type of cancellation: (√(x+4)−2)/x = 1/(√(x+4)+2), for x≠0 in the domain. The limit at 0 is now 1/4.

Continuity, repair and intermediate values

A function is continuous at a when f(a) is defined, the appropriate limit exists and that limit equals f(a). These are three distinct checks. A removable discontinuity occurs when the finite limit exists, but the point was omitted or assigned another value. We can construct a continuous extension by assigning exactly the limit's value. A jump cannot be repaired by changing a single point, since the one-sided behaviours already disagree.

Continuity on an interval permits a powerful conclusion: between two outputs f(a) and f(b), the function takes every intermediate value if it is continuous on [a,b]. Thus, opposite signs guarantee at least one root between the endpoints. They do not guarantee a unique root or automatically provide its exact position. A bisection process reduces the interval while preserving the sign change and produces justified approximations.

Unbounded growth and responsible interpretation

In 1/x², outputs grow without bound as x approaches 0 from either side. We write an infinite limit, +∞, to describe this growth. Infinity is not a real number that can be inserted into the expression. In 1/x, the right side tends to +∞ and the left to −∞; no common two-sided limit exists, even in the extended sense. A vertical asymptote describes unbounded growth on at least one side.

There are also functions that oscillate without approaching an output: sin(1/x) near 0 is an example. A table or calculator may hide these oscillations. When explaining a solution, combine the domain, one-sided approaches and legitimate algebraic transformations. Indicate whether the result is a number, unbounded behaviour or absence of a limit. This classification prepares for studying the derivative, whose initial quotient also often presents 0/0.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

A limit exists only if the function is defined at the point.

A limit uses nearby, distinct inputs; continuity is what also requires a value at the point.

0/0 equals zero.

It is an indeterminate form: different functions can produce different limits or no limit.

A table with five values proves the limit.

Numerical values suggest; a justification must control all nearby behaviour.