Question

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then (A) \(a=1, b=4\) (B) \(a=1, b=-4\) (C) \(a=2, b=-3\) (D) \(a=2, b=3\)

Short answer

The correct answers are a=1 and b=-4, making option (B) the right choice.

Step-by-step solution

Simplify the given expression

Simplify the expression inside the limit by dividing each term of the numerator by the term in the denominator that has the highest power of x, which is x in this case.

Apply the limit

After the simplification, apply the limit as x approaches infinity. This involves identifying terms that approach zero and those that approach a constant value.

Set the resulting expression equal to 4

After applying the limit, the expression will simplify to a constant. This constant should then be set equal to 4, as given in the exercise.

Solve for a and b

Solve the resulting equations from the previous step to find the values of a and b.

Key concepts

Limits at Infinity

In calculus, the concept of \textbf{limits at infinity} refers to the behavior of functions as the input values become very large or very small (approach positive or negative infinity). When faced with a problem like \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), the first step is to understand how each component of the function behaves as \(x\) grows without bound.

To solve these types of problems, we apply certain rules, such as the fact that the ratio of two polynomials will behave like the ratio of their leading terms at infinity. For instance, if a polynomial in the numerator has a higher degree than the denominator, the limit at infinity will often be infinity, if they have the same degree, the limit will be the ratio of the leading coefficients.

In the exercise given, by comparing the degrees of the polynomials, we can anticipate the limit and then proceed to simplify the expression to find the specific values of \(a\) and \(b\) that satisfy the equation. This approach simplifies the problem and allows for the identification of the limit at infinity.

Algebraic Simplification

The step of \textbf{algebraic simplification} is crucial for solving limit problems efficiently. In our original exercise, this involves dividing each term in the numerator by \(x\), the term in the denominator with the highest power. This step effectively reduces the complexity of the function and thus makes the application of the limit more straightforward.

Besides division, algebraic simplification may include factoring, expanding polynomials, or canceling out common factors. By simplifying the expression \(\frac{x^{2}+x+1}{x+1}\), we can cancel terms and identify those that become negligible as \(x\) approaches infinity. This leads us to a form where the behavior of the function as \(x\) goes to infinity is clearer, giving us a simpler equation to work with when taking the limit.

Solving Equations

Once the expression is simplified and the limit is applied, we frequently end up with an equation to solve. This is the last step of our problem-solving process, referred to here as \textbf{solving equations}. After applying the limit and simplifying, we must equate the resultant expression to 4, which is given in the exercise.

Solving for \(a\) and \(b\) involves isolating these variables and systematically determining their values. In solving such equations, one might use various algebraic methods like substitution, elimination, or comparison. The importance lies in the careful manipulation of the equation to solve for the unknowns without altering the equality. For the given exercise, after simplification and applying the limit, we compare coefficients on both sides of the equation to find the values of \(a\) and \(b\) that match the limit value of 4.