limx→ 3(x + 3)

Exercise 12.1 · Q1

\lim_{x\to 3}(x + 3)

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Concept understanding — Limit Of Polynomial

What Happens to a Polynomial as $x$ Approaches a Number?

A limit answers a simple question about a polynomial: as $x$ gets closer and closer to some number $a$ , what value does the polynomial settle near?

The Intuition

Take $P(x) = 3x^2 - 2x + 1$ . What happens as $x$ gets really close to $2$ ?

At $x = 2$ , the polynomial gives $3(4) - 4 + 1 = 9$ .
At $x = 1.9$ , it gives about $8.63$ .
At $x = 2.1$ , it gives about $9.43$ .

The closer $x$ gets to $2$ , the closer $P(x)$ gets to $9$ . There is no drama — the polynomial just slides smoothly to that value.

Note

For any polynomial, $\lim_{x \to a} P(x) = P(a)$ . You can simply substitute the number.

The Precise Statement

\lim_{x \to a} P(x) = P(a)

Why this works. Using the algebra of limits, the limit of a sum is the sum of the limits, and the limit of a constant multiple is the constant times the limit. Since $\lim_{x \to a} x = a$ and $\lim_{x \to a} c = c$ , each term $c_k x^k$ tends to $c_k a^k$ . Adding the terms back together gives exactly $P(a)$ .

A Concrete Example

Find $\lim_{x \to 3} (2x^3 - 5x + 4)$ .

Step 1: Recognise it is a polynomial.

Step 2: Substitute $x = 3$ :

2(27) - 5(3) + 4 = 54 - 15 + 4 = 43

Tip

For polynomials, direct substitution is the only tool you need — no factoring, no rationalising. Just plug in and compute.

The One Trap: "But What If I Can't Plug In?"

You might wonder whether a polynomial can have a hole. It cannot — a polynomial is defined for every real number, so there is never a value you must avoid. The only time "just plug in" can fail is with a rational function (a polynomial divided by another polynomial), where the denominator might be zero. For a pure polynomial, the limit is always the value.

Why This Matters

Limits of polynomials are the foundation for:

Derivatives (the slope of a curve at a point),
Evaluating more complicated limits by simplifying to a polynomial first,
Solving problems in physics such as instantaneous velocity.

This is the simplest, most predictable limit in the whole chapter — everything else builds on it.

Limit of a Polynomial is one of the earliest results in the NCERT Class 11 Mathematics chapter on Limits and Derivatives, matching searches like "limits of polynomial functions formula" or "limits and derivatives important questions class 11". Because the direct-substitution rule is so reliable, it is a quick-scoring question type in both CBSE boards and JEE Main's calculus section.

The limit of a polynomial as $x$ approaches a point is simply the polynomial evaluated at that point. For $\lim_{x\to 3}(x+3)$ , substituting $x=3$ gives $3+3=6$ .

The core idea here is the Limit of a Polynomial property. Polynomials are continuous functions — their graphs have no holes, jumps, or breaks. Because of this, the limit as $x$ approaches any number is just the value of the polynomial at that number. No factoring, no cancellation, no tricks.

Let’s walk through it.

Identify the function.

We have $f(x) = x + 3$ , which is a linear polynomial. It’s defined and continuous for all real $x$ , including $x = 3$ .
Apply the direct substitution property.

For any polynomial $P(x)$ , $\lim_{x \to a} P(x) = P(a)$ . This is a theorem you can rely on in exams. So here:

\lim_{x \to 3} (x + 3) = 3 + 3.

Compute the arithmetic. $3 + 3 = 6$ .

Watch out

A common mistake is to think limits are always about “approaching but never reaching.” That’s true for functions with holes (like $\frac{x^2-9}{x-3}$ ), but for polynomials, the limit equals the function value. Don’t overcomplicate it.

Tip

If you ever forget the polynomial limit rule, just test a few numbers near $x=3$ : try $x=2.9$ gives $5.9$ , $x=3.1$ gives $6.1$ . They all get close to $6$ . That’s the limit.

✓Final answer

\boxed{6}

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Q.lim⁡x→3(x+3)\lim_{x\to 3}(x + 3)limx→3​(x+3)

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Q. $\lim_{x\to 3}(x + 3)$

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