$\lim_{x o \pi}\left(x - \dfrac{22}{7} ight)$

Question

$\lim_{x	o \pi}\left(x - \dfrac{22}{7}ight)$

Accepted Answer

> [!TLDR]
> The limit of a polynomial function as $x$ approaches a constant is simply the polynomial evaluated at that constant. For $\lim_{x	o \pi}\left(x - \frac{22}{7}ight)$, the value is $\pi - \frac{22}{7}$.

The core idea here is that **polynomials are continuous functions**. Continuity means that as $x$ gets arbitrarily close to a point, the function’s value gets arbitrarily close to the function’s value *at* that point. So, for any polynomial $P(x)$, we have:

$$\lim_{x 	o a} P(x) = P(a)$$

This is the “direct substitution” property. It works because polynomials have no breaks, jumps, or holes — they are smooth curves you can trace without lifting your pen.

The expression $x - \frac{22}{7}$ is a linear polynomial (degree 1). Linear polynomials are the simplest continuous functions. So, to find the limit as $x 	o \pi$, we just plug $x = \pi$ into the expression.

Let’s walk through it step by step:

1. **Identify the function type.**  
   $f(x) = x - \frac{22}{7}$ is a polynomial. Specifically, it’s a linear function with slope 1 and y-intercept $-\frac{22}{7}$.

2. **Apply the direct substitution property.**  
   Since polynomials are continuous everywhere, we can evaluate the limit by substituting $x = \pi$ directly:

$$\lim_{x	o \pi}\left(x - \frac{22}{7}ight) = \pi - \frac{22}{7}$$

3. **Interpret the result.**  
   This is not a simplification to a neat number — it’s an exact expression. $\pi$ is an irrational number, and $\frac{22}{7}$ is a rational approximation of $\pi$ (it’s about 3.142857…, while $\pi$ is about 3.14159…). So the limit is a small positive number: $\pi - \frac{22}{7} \approx -0.00126$, which is negative. But the exact answer is left in symbolic form.

> [!WARNING]
> A common mistake is to think $\frac{22}{7}$ equals $\pi$. It does not — $\frac{22}{7}$ is only an approximation. The limit is **not** zero; it’s the exact difference $\pi - \frac{22}{7}$, which is a small negative number.

> [!TIP]
> If you ever see a limit of

Q. $\lim_{x\to \pi}\left(x - \dfrac{22}{7}\right)$

Concept understanding — Limit Of Polynomial

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

The Intuition

The Precise Statement

A Concrete Example

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

Why This Matters

Q.lim⁡x→π(x−227)\lim_{x\to \pi}\left(x - \dfrac{22}{7}\right)limx→π​(x−722​)

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Q. $\lim_{x\to \pi}\left(x - \dfrac{22}{7}\right)$

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