Limits and continuity are key ideas in calculus, a branch of mathematics that studies how things change. A limit tells us what value a function gets closer to as the input (usually called x) gets closer to a certain number. It helps us understand how a function behaves near a point, even if it's not defined exactly at that point.

Continuity means that a function has no breaks, jumps, or holes. If you can draw a function's graph without lifting your pen, the function is continuous. For a function to be continuous at a point, three things must happen: the function must be defined at that point, the limit must exist at that point, and the value of the function and the limit must be the same.

What are Limits and Continuity?

Limits and continuity are important concepts in calculus. A limit helps us understand what value a function is approaching as the input (x) gets close to a certain number (a). If we write it as lim x→a f(x) = l, it means as x gets closer to a, the value of f(x) gets closer to l.

A function is called continuous at a point x = c if three things are true:

1. The function f(c) is defined.
2. The left-hand limit (LHL) and right-hand limit (RHL) of f(x) at x = c both exist.
3. These values are all equal: LHL = RHL = f(c).

LHL means approaching c from the left (smaller values), and RHL means approaching from the right (larger values). Mathematically:

• LHL = lim x→c⁻ f(x) = lim h→0 f(c − h)
• RHL = lim x→c⁺ f(x) = lim h→0 f(c + h)

If both limits give the same result and match the value of the function at x = c, then the function is continuous at that point.

Continuity is a fundamental concept in mathematics that describes the smoothness and unbroken nature of a function or a curve. It refers to the absence of abrupt changes, jumps, or holes in the graph of a function. A function is said to be continuous at a point if it does not have any disruptions or discontinuities at that specific point.

In more technical terms, a function f(x) is continuous at a point c if three conditions are satisfied:

• f(c) is defined (the function must have a value at point c).
• The limit of f(x) as x approaches c exists.
• The limit of f(x) as x approaches c is equal to f(c).

What is the Meaning of Continuity?

Consider any function f(x) which is defined for x = a and is stated to be continuous at x = a, if: f(a) is a finite value.

The limit of the function f(x) as x → a exists and is equal to the value of f(x) at x = a. i.e. \(\lim_{x\to a}f\left(x\right)=l=f\left(a\right)\)

Thus f(x) is continuous at x=a if we have f(a+0)=f(a-0)=f(a), otherwise, the function is discontinuous at x=a.

Types of Discontinuity

• Infinite Discontinuity
• Jump Discontinuity
• Positive Discontinuity

Infinite Discontinuity

Infinite discontinuity is defined as a branch of discontinuity wherever a vertical asymptote is present at x = a, and f(a) is not defined. This is also termed Asymptotic Discontinuity.

Jump Discontinuity

Jump discontinuity is said to occur when; \(\lim _{x \rightarrow a+} f(x) \neq \lim _{x \rightarrow a-} f(x)\) However, both the limits are finite. \(\lim_{x\to a+}f\left(x\right)\ne\lim_{x\to a-}f\left(x\right)\)

Positive Discontinuity

Positive discontinuity occurs when a function has a predefined two-sided limit at x = a, but either f(x) is not defined at a, or its value is not identical to the limit at a.

Definition of Limits

In mathematics, the concept of limits is used to describe the behaviour of a function as its input values approach a certain value. It represents the value that a function approaches or tends to as the input values get arbitrarily close to a particular point. More formally, the limit of a function f(x) as x approaches a (denoted as lim(x→a) f(x)) is defined as the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a, without necessarily reaching that value. The limit provides insights into the overall behaviour of a function, such as its continuity, differentiability, and the existence of asymptotes, and plays a fundamental role in calculus, analysis, and other areas of mathematics.

One-Sided Limit

• A two-sided limit \(\lim _{x \rightarrow a} f(x)\) uses the values of x into a statement that are both greater than and less than a.
• Where’s a one-sided limit from the left \(\lim _{x \rightarrow a-} f(x)\) or from the right \(\lim _{x \rightarrow a+} f(x)\) takes exclusive values of x smaller/greater than a respectively.

Limits and Continuity Formulas

Some useful limits and continuity formulas are given below:

Limit Formulas: Starting with the limit formulas; which cover trigonometric, logarithmic, and exponential followed by the algebra of limits, L’ Hospital’s rule, sandwich theorem and more.

Algebra of Limits: Some of the algebraic formulas regarding limits are as follows:

Sandwich Theorem: Consider f, g, and h to be functions such that f(x) ≤ g(x) ≤ h(x) for all x in some neighbourhood of the point an (except possibly at x = a) and if, then:

\(\lim_{x\to a}f\left(x\right)=l=\lim_{x\to a}h\left(x\right),\ then\ \lim_{x\to a}g\left(x\right)=l\)

Continuity Formulas: If f(a) is defined and \(\lim_{x→a}f(x)\) exists; \(\lim_{x→a^+}f(x)=\lim_{x→a^−}f(x)=f(a)\)

Then a function f is stated to be continuous at the point x = a.

Algebra of continuous functions: If we are given two real functions f and g that are continuous at a real number d. Then the formula below holds good:

• f + g is continuous at x = d.
• f – g is continuous at x = d.
• f . g is continuous at x = d.
• \(\frac{f}{g}\) is continuous at x = d, (when g (d) ≠ 0).

Discontinuity of a Function: A function f(x) which is not continuous at a point x = a, then a function f(x) is said to be discontinuous at x = a.

Difference Between Limits and Continuity

The important difference between Limits and Continuity is given below:

Important Points of Limits and Continuity

• For a function \(f(x)\) the limit of the function at a point \(x=a\) is the value the function attains at a point that is very near to \(x=a\).
• If the limit is defined in terms of a number that is smaller than a: then the limit is called the left-hand limit. It is denoted as \( x \rightarrow a^{-} \)
• If the limit is defined in terms of a number that is greater than a then: the limit is called the right-hand limit. It is denoted as \( x \rightarrow a^{+} \)
• The limit of a function \( f(x) at x=a \) exists: only when it's left-hand limit and right-hand limit exist and are equal and possess a finite value.
  i.e. \( \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x) \)
• A function is said to be continuous over a range: if it′s graph is a single unbroken curve.
• If a function \(f(x)\) is continuous at \(x=x_{\circ}\)
  i.e. \( \lim_{x\to x_{\circ}^+}f(x)=\lim_{x\to x_{\circ}^-}f(x)=\lim_{x\to x_{\circ}}f(x) \)

Properties of Limits 

1. Basic Idea:
   A limit tells us what value a function gets close to as the input (x) gets close to a certain number (a).
   We write this as:
   limₓ→ₐ f(x) = L,
   which means f(x) gets closer to L when x gets closer to a.

2. Sum Rule:
   If you're adding two functions, the limit of their sum is just the sum of their limits:
   limₓ→ₐ [f(x) + g(x)] = limₓ→ₐ f(x) + limₓ→ₐ g(x)

3. Constant Rule:
   The limit of a constant is just the constant itself:
   limₓ→ₐ C = C

4. Constant Multiple Rule:
   If a function is multiplied by a number (say m), you can take the limit of the function first and then multiply:
   limₓ→ₐ [m × f(x)] = m × limₓ→ₐ f(x)

5. Quotient Rule:
   For dividing two functions, the limit of the division is the division of their limits — as long as the denominator’s limit isn’t zero:
   limₓ→ₐ [f(x)/g(x)] = limₓ→ₐ f(x) / limₓ→ₐ g(x) (if the bottom part ≠ 0)

Solved Examples of Limits and Continuity

With all the knowledge of limits followed by continuity including definitions, formulas, types and key takeaways it’s time to practice some examples for more clarity:

Example 1: Find the limits for the expression given by \(\lim_{x\to1}\left[2x^3-3x^2+2\right]\)

Solution: Given function \(\lim_{x\to1}\left[2x^3-3x^2+2\right]\) substitute the limits.

Therefore \(\lim_{x\to1}\left[2x^3-3x^2+2\right]\) =1

Example2: Evaluating the limit for the function;\(\lim_{x\to1}\ \frac{x\left(x-2\right)^2}{\left(x^2-4\right)}\).

Solution: Given function is \(\lim_{x\to1}\ \frac{x\left(x-2\right)^2}{\left(x^2-4\right)}\).

Open the function into its component as shown:

\(\lim_{x\to1}\ \frac{x\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}\)

Now substituting the limits we get:

\(\frac{x\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}=\frac{x\left(x-2\right)}{\left(x+2\right)}=\frac{1\left(1-2\right)}{\left(1+2\right)}=\frac{-1}{3}\)

Example 3: Check for the continuity of the function f given by the expression f (x) = 4x + 5 at x = 1.

Solution: First check for the limit:

\(\lim_{x\to1}\ f\left(x\right)=\lim_{x\to1}\ \left(4x+5\right)=9\)

Now check for f(1)

f (x) = 4x + 5

f (1) = 4(1) + 5=9

Hence \(\lim_{x\to1}\ f\left(x\right)=9=f\left(1\right)\)

Therefore, the given function is continuous at x = 1.

Through this article, we learned about limits and continuity as a part of mathematical calculus and explored concepts of limits via existence, properties, indeterminate form, and Sandwich Theorem followed by Exponential, Logarithmic, and Trigonometric limits in succession with continuity by covering topics like the continuity of a function in the interval, properties, and types of discontinuity along with key ideas of Infinitesimals.

Stay tuned to the Testbook app or visit the Testbook website for more updates on similar topics from mathematics, science, and numerous such subjects, and can even check the test series available to test your knowledge regarding various exams.