Q.$\lim_{x\to 0}\dfrac{ax + b}{cx + 1}$

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.

The Precise Statement

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 $x = 3$:

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