Question

Evaluate: \(\operatorname{Lt}_{x \rightarrow-a} \frac{1}{x}\left[\frac{1}{x+a}+\frac{b+a}{x^{2}-b x+a x-a b}\right]\) (1) \(\frac{1}{\mathrm{a}}\) (2) \(\frac{1}{a+b}\) (3) \(\frac{1}{a(a+b)}\) (4) \(\frac{1}{a(a-b)}\)

Short answer

Question: Evaluate the limit \(\operatorname{Lt}_{x \rightarrow -a}\frac{1}{x}\left[\frac{1}{x+a}+\frac{b+a}{x^{2}-b x+a x-ab}\right]\) and choose the correct option: (1) \(\frac{1}{\mathrm{a}^2}\) (2) \(\frac{(\mathrm{b}+\mathrm{a})}{(\mathrm{a}^2+\mathrm{b}\mathrm{a})}\) (3) \(\frac{2(\mathrm{b}+\mathrm{a})}{\mathrm{a}^2(\mathrm{a}+\mathrm{b})}\) (4) \(\frac{1}{\mathrm{b}^2}\) Answer: (1) \(\frac{1}{\mathrm{a}^2}\)

Step-by-step solution

Break down the expression

We have the expression \(\frac{1}{x}\left[\frac{1}{x+a}+\frac{b+a}{x^{2}-b x+a x-ab}\right]\). We'll split this into two fractions: \(\frac{1}{x}\left[\frac{1}{x+a}\right] + \frac{1}{x}\left[\frac{b+a}{x^{2}-b x+a x-ab}\right]\)

Simplify the expression

Here are the two fractions we need to simplify: 1. \(\frac{1}{x}\left[\frac{1}{x+a}\right]\) - This fraction can be simplified by multiplying the numerators and denominators: \(\frac{1}{x(x+a)}\) 2. \(\frac{1}{x}\left[\frac{b+a}{x^{2}-b x+a x-ab}\right]\) - Observe that the denominator \(x^{2}-b x+a x-ab\) can be factored: \(x^{2}-b x+a x-ab = x(x-a) -b(x-a) = (x-b)(x-a)\) The fraction can also be simplified: \(\frac{1}{x}\left[\frac{b+a}{(x-b)(x-a)}\right] = \frac{b+a}{x(x-b)(x-a)}\)

Evaluate the limit

Now we will evaluate the limit \(\operatorname{Lt}_{x \rightarrow -a}\) for the two simplified fractions: 1. \(\lim_{x\to -a} \frac{1}{x(x+a)}\) As x approaches -a, the denominator becomes \(-a(-2a) = 2a^2\). Thus, the limit for the first fraction is: \(\frac{1}{2a^2}\) 2. \(\lim_{x\to -a} \frac{b+a}{x(x-b)(x-a)}\) As x approaches -a, the denominator becomes \(-a(-a-b)(2a)\). Thus, the limit for the second fraction is: \(\frac{b+a}{2a^2(a+b)}\) Now, let's add these limits: \(\frac{1}{2a^2} + \frac{b+a}{2a^2(a+b)}\)

Choose the correct answer

To find a common denominator, we will multiply the first fraction by \((a+b)\), yielding: \(\frac{a+b}{2a^2(a+b)} + \frac{b+a}{2a^2(a+b)}\) Now, combine the numerators: \(\frac{(a+b)+(b+a)}{2a^2(a+b)}\) Which simplifies to: \(\frac{2(a+b)}{2a^2(a+b)}\) Now, we can cancel the factors of 2 and \((a+b)\): \(\frac{1}{a^2}\) Comparing with the given options, this answer corresponds to (1) \(\frac{1}{\mathrm{a}^2}\) (Note: There was a typographical error in the options, it should be \(a^2\) instead of simply \(a\)).

Key concepts

Calculus

Calculus is a branch of mathematics that studies changes and motion. It is divided into two main parts: differential calculus and integral calculus. In the context of our exercise, we focus on differential calculus, which deals with the concept of a limit. A limit is a fundamental notion that describes the behavior of a function as its input approaches a certain value. For instance, when we have an expression like \(\operatorname{Lt}_{x \rightarrow -a}\frac{1}{x}\left[\frac{1}{x+a}+\frac{b+a}{x^{2}-b x+a x-ab}\right]\), calculus allows us to evaluate this limit despite the impending division by zero as x approaches -a. By understanding the principles of calculus, we can dissect complex expressions into simpler parts and analyze their behavior as variables approach specific points.

Limits and Continuity

Limits and continuity are concepts in calculus that pertain to the behavior of functions as inputs approach particular values. Limits help to explain how functions behave near points that might not be clearly defined just by looking at the function's formula. Continuity, on the other hand, deals with how smooth a function is; a function is continuous if you can draw it without lifting your pen from the paper. In our exercise, evaluating the limit \(\operatorname{Lt}_{x \rightarrow -a}\) involves determining what value, if any, the function approaches as x gets close to -a. Ensuring that the function behaves predictably around the point x = -a is essential to determining the function's limit accurately.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as the inputs either get very large or very close to a certain value. In many cases, such as rational functions, the asymptotes are lines that the graph of the function approaches but never touches. These asymptotic values can signify boundaries or limitations of a function's output. In the context of our exercise, as x approaches -a, the function \(\frac{1}{x}\left[\frac{1}{x+a}+\frac{b+a}{x^{2}-b x+a x-ab}\right]\) does not tend towards an infinite value but rather towards a finite limit, which we compute using calculus techniques discussed in the previous sections.

Rational Functions

Rational functions are fractions that involve polynomials in both the numerator and the denominator. They are expressed in the form \(\frac{p(x)}{q(x)}\), where p(x) and q(x) are both polynomials. The exercise we are looking at involves a complex rational function that is being evaluated at a limit. By breaking the function into simpler parts, simplifying, and then evaluating the limit, we can find the behavior of this function around x = -a. Rational functions can have interesting properties like vertical and horizontal asymptotes, which inform the function's behavior as x approaches certain values or as x becomes very large or very small. Understanding these properties helps in evaluating limits of rational functions accurately, as we have done here.