Problem 132 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\lim _{x \rightarrow-1} \frac{x^{13}+1}{x^{17}+1} \text { \\{Ans. } \frac{13}{17}\right\\} $$

Step-by-Step Solution

Answer

The short answer for the limit of the given function as x approaches -1 is: \[\lim_{x\to -1} \frac{x^{13}+1}{x^{17}+1} = 1\]

Step 1: 1: Analyze the function for simplification

Notice that we have a common factor in the numerator and denominator. We can rewrite the given function as:\[\frac{x^{13} + 1}{x^{17} + 1} = \frac{(x^{13} + 1)(x^4 - x^4)}{(x^{17} + 1)(x^4 - x^4)}\] By applying the difference of squares identity, we can further simplify the expression to:\[\frac{(x^{13} + 1)((x^{(4/2)} - x^{(0)})(x^{(4/2)} + x^{(0)}))}{(x^{17} + 1)((x^{(4/2)} - x^{(0)})(x^{(4/2)} + x^{(0)}))}\]

Step 2: 2: Cancel common terms

The expression we derived above from the given function contains common factors in the numerator and denominator, which we can cancel:\[\frac{x^{13} + 1}{x^{17} + 1} = \frac{(x^{13} + 1)(x^2 - x^0)}{(x^{17} + 1)(x^2 - x^0)} = \frac{x^{13} + 1}{x^{17} + 1}\]

Step 3: 3: Apply the limits

Now, we can apply the limit as x approaches -1:\[\lim_{x\to -1} \frac{x^{13}+1}{x^{17}+1}\]

Step 4: 4: Evaluate the limit

As x approaches -1, we have:\[\frac{(-1)^{13}+1}{(-1)^{17}+1} = \frac{-2}{-2}\]

Step 5: 5: Simplify the limit

Therefore, the limit simplifies to:\[\frac{-2}{-2} = 1\] In conclusion, the limit of the given function as x approaches -1 is equal to 1.

Key Concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of how functions change, focusing chiefly on concepts of derivatives and limits. The derivative is a key concept here, representing the rate of change with respect to a variable. It's akin to understanding the slope of a curve at a specific point. This makes differential calculus foundational in fields like physics and engineering, where it's necessary to model and predict continuous change.

In our particular problem, we sought the limit of a function as it approaches a certain point, which is closely tied to the derivative. Understanding limits is vital, as they often serve as the building blocks for defining derivatives. Once the behavior of a function at a specific point is understood through limits, it opens the door to defining and calculating the rate of change. Consider it as preparing the ground before understanding how steep our path is at a point.

Limit Evaluation

Limits are fundamental tools in calculus that help us understand the behavior of functions as inputs approach a certain value. In this problem, we evaluated the limit of a complex fraction as the variable approached -1. The core idea is to figure out what value the function is teasing towards, even if it never quite reaches that point.

One of the ways to solve limits is by simplifying complex expressions. Here, we canceled common terms through algebraic manipulation, but limits can also be solved by direct substitution or advanced techniques like L'Hopital's rule.

• The key is recognizing if the direct substitution leads to an indeterminate form, such as \( \frac{0}{0} \), and then employing algebraic tricks to resolve it.
• For cases like the one in the exercise, checking the function's behavior at points near the limit might also provide insights.

Evaluating limits is a pathway to deeper insight into function behavior, helping anticipate trends such as approaching vertical asymptotes or horizontal levels.

Polynomial Functions

Polynomial functions like the one in our exercise are expressions made up of variables raised to integer powers and their coefficients. They are some of the simplest yet most versatile types of functions in calculus. The given expression \( \frac{x^{13} + 1}{x^{17} + 1} \) demonstrates a polynomial function's capabilities to model complex behaviors, such as asymptotic behavior and leading terms that significantly impact the limit evaluation.

Polynomial functions are handy because:

• They are smooth and continuous, meaning no sudden jumps or breaks.
• They are closed under addition, subtraction, and multiplication operations, making them convenient to manipulate algebraically.
• They form the basis for building more complex functions through addition and multiplication with coefficients.

The limit we evaluated depended heavily on understanding how these functions behave as their input values grow or shrink to extremes, like approaching -1. Identifying leading terms and simplifying expressions usually provides the limit's answer, owing to these functions' propensity to approach known patterns.