Problem 72 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 2} \frac{x^{3}+3 x^{2}-9 x-2}{x^{3}-x-6}\left\\{\text { Ans. } \frac{15}{11}\right\\} $$

Step-by-Step Solution

Answer

The short answer based on the provided step-by-step solution is: The correct limit of the function \(\lim\limits_{x\to 2}\frac{x^3 + 3x^2 - 9x - 2}{x^3 - x - 6}\) is \(5\), and the given answer \(\frac{15}{11}\) is incorrect.

Step 1: Identify the given limit

The limit we want to find is \(\lim\limits_{x\to 2}\frac{x^3 + 3x^2 - 9x - 2}{x^3 - x - 6}\).

Step 2: Substitute x = 2 into the given function

When x = 2, we need to find the value of the function to determine if it is an indeterminate form or not. \(f(2) = \frac{(2)^3 + 3(2)^2 - 9(2) - 2}{(2)^3 - (2) - 6}\)

Step 3: Calculate the expression with x = 2

After substituting x = 2, the expression becomes: \[\frac{8 + 12 - 18 - 2}{8 - 2 - 6}\] Simplify the expression: \[\frac{15}{3}\]

Step 4: Identify the Limit

Since the function is not in an indeterminate form, we can directly say that: \[\lim\limits_{x\to 2}\frac{x^3 + 3x^2 - 9x - 2}{x^3 - x - 6} = \frac{15}{3} = 5\] The given answer \(\frac{15}{11}\) is incorrect. The correct answer is 5.

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, the given function in the exercise is a ratio of two polynomials:

• Numerator: \(x^3 + 3x^2 - 9x - 2\)
• Denominator: \(x^3 - x - 6\)

Both expressions are cubic polynomials, meaning their highest degree term is \(x^3\). Understanding the structure of polynomials is crucial for evaluating limits. Polynomials are continuous everywhere, which allows us to apply certain techniques easily when evaluating limits. Keep in mind, for polynomials:

• The degree of a polynomial is given by the highest power of the variable in the polynomial.
• If polynomials are arranged in standard form, it starts with the term of highest degree.

When dealing with polynomials in the context of limits, a common goal is to simplify the expression to avoid any indeterminate forms.

Indeterminate Forms

When evaluating limits, certain expressions can lead to indeterminate forms. These forms occur when a straightforward substitution yields expressions like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or others.
In such cases, the limit cannot be directly determined without further simplification or analysis.
In this exercise, when substituting \(x = 2\) into the expression, we check if it results in an indeterminate form. If it did, it would be necessary to apply algebraic manipulations such as factoring or using L'Hopital's Rule to resolve this indeterminacy.

• Indeterminate forms suggest that a more sophisticated approach is necessary to evaluate the limit.
• Identifying indeterminate forms is crucial as it guides us in choosing the right technique to evaluate the limit.

Fortunately, in our exercise, the substitution did not lead to an indeterminate form, hence simplifying our calculation process.

Direct Substitution

Direct substitution is a technique used to evaluate limits, particularly when dealing with continuous functions such as polynomials. It involves substituting the value that \(x\) approaches directly into the function.
In the given exercise, since both the numerator and the denominator are polynomials, we first attempt direct substitution to see if it yields a valid result.

• If the result is not an indeterminate form, the limit can be evaluated directly.
• Direct substitution is typically the first method applied, and if it fails, other techniques are considered.

In our problem, substituting \(x = 2\) gave us a straightforward answer:\[f(2) = \frac{15}{3} = 5\]This confirms the simplicity and effectiveness of direct substitution when applicable. Once the expression is simplified and resolved, confirming with the initial conditions ensures accuracy in determining limits.