Problem 48 - Precalculus with Limits

Question

In Exercises 47 and 48, find $$\textrm{(a)}\ \lim_{x \to 2}\ f(x), \quad \textrm{(b)} \lim_{x \to 2}\ g(x), \quad \textrm{(c)} \lim_{x \to 2}\ [f(x)g(x)], \quad \textrm{and (d)} \lim_{x \to 2}\ [g(x)-f(x)].$$ $$f(x)=\dfrac{x}{3-x}, \quad \quad g(x)=\sin \pi x$$

Step-by-Step Solution

Answer

Therefore, \(\lim_{x \to 2}\ f(x) = 2\), \(\lim_{x \to 2}\ g(x) = 0\), \(\lim_{x \to 2}\ [f(x)g(x)] = 0\) and \(\lim_{x \to 2}\ [g(x)-f(x)] = -2\)

Step 1: Find the limit of f(x) as x approaches 2

Substitute x with 2 in the function f(x) = x/(3-x), \n Resulting in f(2) = 2/(3-2) which is equal to 2.

Step 2: Find the limit of g(x) as x approaches 2

Substitute x with 2 in the function g(x) = sin(\(\pi x\)), \n Resulting in g(2) = sin(\(\pi * 2\)) which is equal to 0 as sin(\(2 \pi \)) is always equals to 0.

Step 3: Find the limit of f(x)g(x) as x approaches 2

Multiplication of the two functions f(x) and g(x) at x=2, \n Resulting in f(2)g(2) = 2*0 which is equal to 0.

Step 4: Find the limit of g(x)-f(x) as x approaches 2

Subtraction of the two functions f(x) and g(x) at x=2, \n Resulting in g(2)-f(2) = 0-2 which is equal to -2.

Key Concepts

Function Operations

Function operations involve combining functions using arithmetic operations such as addition, subtraction, multiplication, and division. This exercise asks you to compute limits using these operations. When calculating limits, remember you're finding the behavior of a function as it approaches a certain point without necessarily reaching that point.

• For instance, in this exercise, examining the limit of \((f(x)g(x))\) involves multiplying individual limits. If these limits exist, \((\lim_{{x \to 2}} f(x)g(x) = \lim_{{x \to 2}} f(x) \cdot \lim_{{x \to 2}} g(x))\), provided neither is infinite or undefined.
• Similarly, the limit of a difference, such as \((g(x) - f(x))\), corresponds to subtracting limits: \((\lim_{{x \to 2}} g(x) - \lim_{{x \to 2}} f(x))\).

Understanding these operations is crucial to evaluating limits of composite expressions.

Trigonometric Functions

Trigonometric functions are fundamental to understanding waves, circles, and oscillations, and they appear frequently in calculus problems. In this exercise, we have a trigonometric function \((g(x) = \sin(\pi x))\).

• The sine function, \(\sin(\theta)\), represents the y-coordinate of a point on the unit circle for an angle \(\theta\). It oscillates between -1 and 1.
• At specific important angles, like \(0, \pi, 2\pi\), the sine function equals 0. This is fundamental when calculating the limit, as \(\sin(2\pi) = 0\).

Recognizing these properties allows you to simplify calculations and immediately understand behaviors of trigonometric functions in limit problems.

Evaluating Limits

Evaluating limits requires carefully analyzing how functions behave as input values approach a point. This can be done via substitution, factoring, or recognizing special function behavior.

• Substitution is the most straightforward method, used when the function is defined and well-behaved at the target point. If direct substitution leads to an undefined form like \(\frac{0}{0}\), other techniques may be necessary.
• In this exercise, substitution worked effectively, allowing us to compute \(\lim_{{x \to 2}} f(x)\) and \(\lim_{{x \to 2}} g(x)\).

Mastery of limits is foundational, as it serves as a stepping stone to more advanced calculus topics like differentiation and integration.

Rational Functions

Rational functions are fractions involving polynomials, such as \(f(x) = \frac{x}{3-x}\). They are characterized by the presence of a variable in the denominator.

• When evaluating limits, always check where the denominator can become zero, as this indicates potential vertical asymptotes or undefined points.
• With \(f(x)\), substitution yields \(2/(3-2) = 2\), since the function is not approaching a problematic undefined form at \(x = 2\).

Understanding the behavior of these functions enhances your ability to predict outcomes in limits and identify asymptotic behavior or potential discontinuities.