Animated Solution for Mathematics - Limits, Continuity and Differentiability: If ${\lim }_{x\to \infty }\left(\frac{x^2+x+1}{x+1}-ax-b\right)=4$, then

Imagine you are standing on a vast, infinite plane, watching a curve defined by $f(x)=\frac{x^2+x+1}{x+1}$. As $x$ grows larger and larger, this curve begins to settle into a predictable pattern, mimicking a straight line known as an oblique asymptote.

Our mission is to uncover the secret identity of this line, defined by $y=ax+b$, such that the gap between our curve and this line remains exactly $4$ as $x$ approaches infinity.

When you see a rational function like $f(x)=\frac{x^2+x+1}{x+1}$, you can perform polynomial division to reveal the structure hidden beneath. We rewrite the numerator $x^2+x+1$ as $x(x+1)+1$.

Now, watch the magic happen:

Suddenly, the complexity vanishes. We see that for very large $x$, the term $\frac{1}{x+1}$ becomes negligible, and the function behaves exactly like the line $y=x$.

We are given the condition that the limit of the difference between the function and the line is $4$:

Substituting our simplified form of $f(x)$, we get:

Let's group the terms by their power of $x$:

If the coefficient $(1-a)$ were anything other than zero, the entire expression would explode to infinity as $x$ grows. Because the problem guarantees a finite limit of $4$, we must force the coefficient of $x$ to be zero.

This gives us our first breakthrough: $1-a=0$, which implies $a=1$.

With $a=1$ locked in, the limit expression simplifies significantly as the $x$ terms cancel out:

As $x$ marches toward infinity, the term $\frac{1}{x+1}$ gracefully fades to zero. We are left with the simple, elegant equation $-b=4$.

This tells us that $b=-4$.

We have successfully determined that the curve $f(x)$ approaches the line $y=x-4$ with a constant vertical offset of $4$. This result demonstrates how calculus allows us to see the underlying order in what initially appears to be a complex, shifting function.