Question

In the following cases, find $limf(x)$ in two ways: (i) use definitions only; (ii) use suitable theorems and justify each step accordingly. (a) $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}(=0)$. (b) $\underset{x\to \mathrm{\infty }}{lim}\frac{x(x-1)}{1-3x^2}$. (c) $\underset{x\to 2^{+}}{lim}\frac{x^2-2x+1}{x^2-3x+2}$. (d) $\underset{x\to 2^{-}}{lim}\frac{x^2-2x+1}{x^2-3x+2}$. (e) $\underset{x\to 2}{lim}\frac{x^2-2x+1}{x^2-3x+2}(=\mathrm{\infty })$.

Short answer

(a) 0, (b) -1/3, (c) ∞, (d) -∞, (e) ∞.

Step-by-step solution

Part (a) - Using Definitions

To find $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}$, consider the function $f(x)=\frac{1}{x}$. By definition, $\underset{x\to \mathrm{\infty }}{lim}f(x)=L$ iff for every $\epsilon >0$ there exists a $N>0$ such that for all $x>N$, $|f(x)-L|<\epsilon$. Here, $|\frac{1}{x}|=\frac{1}{x}<\epsilon$ when $x>\frac{1}{\epsilon }$. Thus, as $x\to \mathrm{\infty }$, $f(x)=\frac{1}{x}\to 0$.

Part (a) - Using Theorem

Using the theorem about limits at infinity, we know that if $f(x)=\frac{1}{x}$ and $x\to \mathrm{\infty }$, then $\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}=0$ because $\frac{1}{x}\to 0$ as $x\to \mathrm{\infty }$. The function approaches zero because the denominator $x$ becomes infinitely large.

Part (b) - Using Definitions

Given $f(x)=\frac{x(x-1)}{1-3x^2}$, we need to find the limit as $x\to \mathrm{\infty }$. Simplify: $$f(x)=\frac{x^2-x}{1-3x^2}=\frac{-1}{3}(1-\frac{1}{x})$$. As $x\to \mathrm{\infty }$, $\frac{1}{x}\to 0$, so $f(x)\to \frac{-1}{3}$. Thus, $\underset{x\to \mathrm{\infty }}{lim}f(x)=\frac{-1}{3}$.

Part (b) - Using Theorems

Applying the theorem about limits involving rational functions: When the degree of the polynomial in the numerator (2 here) is equal to the degree of the polynomial in the denominator (2 here), the limit is the ratio of the leading coefficients. Hence, $\underset{x\to \mathrm{\infty }}{lim}\frac{x(x-1)}{1-3x^2}=\frac{1}{-3}=\frac{-1}{3}$.

Part (c) - Using Definitions

For $\underset{x\to 2^{+}}{lim}\frac{x^2-2x+1}{x^2-3x+2}$, the expression simplifies to $\frac{(x-1{)}^2}{(x-2)(x-1)}$. As $x\to 2^{+}$, it becomes $\frac{1}{0^{+}}$ leading to infinity.

Part (c) - Using Theorems

Notice that as $x\to 2^{+}$, the zero in the denominator is of first degree, causing a vertical asymptote. Thus, by theorem, $\underset{x\to 2^{+}}{lim}\frac{x^2-2x+1}{x^2-3x+2}=\mathrm{\infty }$.

Part (d) - Using Definitions

For $\underset{x\to 2^{-}}{lim}\frac{x^2-2x+1}{x^2-3x+2}$, it simplifies the same as in part (c). As $x\to 2^{-}$, it becomes $\frac{1}{0^{-}}$, tending to $-\mathrm{\infty }$.

Part (d) - Using Theorems

By the theorem for limits approaching vertical asymptotes, $\underset{x\to 2^{-}}{lim}\frac{x^2-2x+1}{x^2-3x+2}=-\mathrm{\infty }$, due to the negative small denominator.

Part (e) - Using Definitions

For $\underset{x\to 2}{lim}\frac{x^2-2x+1}{x^2-3x+2}$, there is a zero in the denominator where $x=2$. The expression becomes $\frac{1}{0}$, indicating an undefined form leading to infinity due to limits from both sides.

Part (e) - Using Theorems

According to the theorem, as $x\to 2$ from either side, the denominator approaches zero, creating a discontinuity. Hence, $\underset{x\to 2}{lim}\frac{x^2-2x+1}{x^2-3x+2}=\mathrm{\infty }$ due to the presence of a vertical asymptote at $x=2$.

Key concepts

Definitions in Calculus

Calculus is a branch of mathematics that deals with change and motion. In particular, limits play an essential role. The limit of a function describes how the function behaves as its input approaches a certain value.

To find a limit using definitions, we focus on the concept of the epsilon-delta definition. This states that a function $f(x)$ approaches a limit $L$ as $x$ approaches $c$ if for every positive number $\epsilon$, there exists a positive number $\delta$ such that whenever $0<|x-c|<\delta$, it follows that $|f(x)-L|<\epsilon$.

This definition is fundamental in calculus as it provides a precise mathematical basis for the concept of a limit, allowing us to handle limits rigorously. Understanding this definition is crucial for tackling more complex functions and behaviors.

Limit Theorems

Limit theorems are powerful tools used to simplify the calculation of limits. A common theorem is the limit of sums, products, and quotients, which helps break down complicated expressions into manageable pieces.

For example, the theorem concerning the limit at infinity states that if $f(x)=\frac{1}{x}$, then $\underset{x\to \mathrm{\infty }}{lim}f(x)=0$, because as $x$ gets larger, $\frac{1}{x}$ shrinks towards zero.

Another useful theorem is that for rational functions. When the degrees of the polynomials in the numerator and denominator are equal, the limit at infinity equals the ratio of their leading coefficients. These theorems help make complex limit problems more approachable and allow for more straightforward calculations.

Vertical Asymptotes

Vertical asymptotes are lines that a graph of a function approaches but never touches or crosses. They occur where the function heads towards infinity as the input approaches a certain finite value.

A vertical asymptote is often found by identifying points where the denominator of a rational function becomes zero, causing division by zero. For example, in a function like $\underset{x\to 2}{lim}\frac{x^2-2x+1}{x^2-3x+2}$, setting the denominator to zero helps locate potential asymptotes, indicating discontinuity and infinite behavior.

This concept is crucial for understanding the behavior of graphs and functions at certain points, especially when computing limits.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They take the form $\frac{p(x)}{q(x)}$ where both $p(x)$ and $q(x)$ are polynomial expressions. Understanding limits involving rational functions often involves investigating points of discontinuity and applying limit theorems.

When dealing with limits of rational functions, it is common to simplify the expression by factoring. This allows us to identify any common factors between the numerator and the denominator which might define the limit behavior or reveal points of indeterminacy, such as zeros in the denominator.

By correctly applying algebraic simplifications and limits theorems, handling rational functions becomes manageable and can lead to calculated predictions of function behaviors, particularly around points of discontinuity.