Question

\(\operatorname{Lt}_{x \rightarrow \infty} \frac{6 x^{4}+7 x^{3}+2 x+1}{x^{4}+1}=\) (1) 1 (2) 2 (3) 3 (4) 6

Short answer

A) 0 B) 2 C) Infinity D) -2 E) Undefined Answer and explanation: 1. Identify the leading terms: In the numerator, the leading term is 4x^6, and in the denominator, the leading term is 2x^4. 2. Divide the leading terms: Divide 4x^6 by 2x^4: (4x^6)/(2x^4) 3. Simplify the expression: When we simplify the expression, we get: (4x^6)/(2x^4) = 2x^2 4. Find the limit: As x approaches infinity, the expression 2x^2 also approaches infinity. 5. Choose the correct answer: Comparing our result to the given options, we find that the correct answer is C) Infinity.

Step-by-step solution

Identify the leading terms

We need to identify the leading terms in the numerator and the denominator. The leading term in the numerator is \(6x^4\), and the leading term in the denominator is \(x^4\).

Divide the leading terms

Divide the leading term in the numerator by the leading term in the denominator: \(\frac{6x^4}{x^4}\)

Simplify the expression

When we simplify the expression, we get: \(\frac{6x^4}{x^4} = 6\)

Find the limit

Now we can find the limit as x approaches infinity: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{6 x^{4}+7 x^{3}+2 x+1}{x^{4}+1} = 6\)

Choose the correct answer

Comparing our result to the given options, we find that the correct answer is (4) 6.

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It's divided into two main areas: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. Integral calculus, on the other hand, involves the integral, which represents the accumulation of quantities and can be used to calculate areas, volumes, and concepts related to total change.

In the context of solving limit problems, calculus provides a systematic way to understand behavior of functions as inputs grow very large (limits at infinity) or approach a particular value (finite limits). The limit concept is foundational in calculus because it underpins the definitions of the derivative and the integral.

Limits at Infinity

Limits at infinity refer to finding the value that a function approaches as the input grows without bound. When evaluating limits at infinity, we're often interested in the end behavior of a polynomial or rational function as the independent variable, typically represented as 'x', approaches positive or negative infinity.

The key technique in solving these limits is to recognize the leading terms in the numerator and denominator, as these terms will dictate the function's behavior at extreme values. For higher degree polynomials, the highest power of 'x' dominates and determines the limit. In our example, the leading term of both the numerator and the denominator is a fourth degree term, which simplifies to a constant as 'x' approaches infinity.

Algebraic Simplification

Algebraic simplification is a process of reducing expressions to their simplest form. It's a crucial step in mathematical problem-solving and often involves factoring, expanding polynomials, canceling common terms, and rationalizing denominators, among other techniques.

Applying Algebraic Simplification to Limits

When dealing with limits, particularly those at infinity, it's common to simplify the expression by dividing by the highest power of 'x' present in the function. This helps to reduce the number of terms and often renders terms with lower powers of 'x' negligible, making it easier to see what value the function approaches. In our example, we simplified the expression by dividing both terms by the highest power of 'x', allowing us to quickly determine the limit as 'x' goes to infinity.

Mathematical Analysis

Mathematical analysis is the branch of mathematics that deals with limits and related theories, such as derivatives and integrals. It encompasses the rigorous study of real and complex number systems, sequences, series, and functions. Analysis lays down the foundations for calculus and provides the necessary proofs to ensure the methods used are sound and reliable.

In the context of solving limit problems, analysis would involve examining the behavior of a function by considering sequences of values that approach a particular point, including infinity. It helps mathematicians to establish the properties of functions and to rigorously justify the operations performed during the problem-solving process. In the exercise demonstrated, analysis would cover why dividing by the highest power of 'x' yields the correct behavior of the rational function at infinity.