1. Explain the concept of infinite limits andhow they relate to vertical asymptotes. 2. How do you evaluate a limit using the Squeeze Theorem?Provide an example. 3. Define continuity for functions andexplainhow it differs from differentiability. 4. What conditions must be satisfied for a function to have a maximum orminimum value ona closed interval according to the Maximum-Minimum Theorem? 5. Describe the relationship betweenthe first derivative and the behavior of a function (increasing. decreasing) using the FirstDerivative : Test 6. How do you determine the left-hand limit and right-hand limit of a function atpointc? 7. How do you find the derivative of a function using the definition of the derivative? 8. For each of the following functions, find f'(c) using the definition a. f(x)=2x-4 at c=1 b. f(x)=x^2+3 at c=-1 C. f(x)=x^3-2 at c=0 d. f(x)=vert x+2vert at c=2 9. A Find the derivative of the following functions a. f(x)=(x^2-5)cosx b. g(x)=sqrt (x)secx C. (2x^3-5x)/(x^2)+3 d. g(x)=(2x)/(tanx) 10. A Sketch the graph of the following functions a) f(x)=(x^2-1)^2 b) g(x)=(e^x)/(x) c) g(x)=(x^3-3x^2+4)/(x^2)-1

1. Infinite limits refer to the behavior of a function as it approaches a particular value from either the left or right side. When a function approaches positive or negative infinity as it approaches a certain value, it is said to have an infinite limit. Infinite limits are related to vertical asymptotes, which are vertical lines on a graph where the function approaches positive or negative infinity as it gets closer to the line.

2. The Squeeze Theorem is a method used to evaluate limits by "squeezing" the function between two other functions whose limits are known. For example, if we have three functions f(x), g(x), and h(x) such that f(x) ≤ g(x) ≤ h(x) for all x in a certain interval, and if the limit of f(x) and h(x) as x approaches a certain value is the same, then the limit of g as x approaches that value is also the same.

3. Continuity of a function refers to the property where the function does not have any breaks, jumps, or holes in its graph. A function is continuous at a point if the limit of the function as it approaches that point exists and is equal to the function's value at that point. Continuity differs from differentiability in that a function can be continuous without being differentiable, but a differentiable function must be continuous.

4. According to the Maximum-Minimum Theorem, for a function to have a maximum or minimum value on a closed interval, the function must be continuous on that interval. Additionally, the function must either be decreasing or increasing on the entire interval, or it must change between increasing and decreasing at most once.

5. The First Derivative Test is a method used to determine the behavior of a function based on the sign of its first If the first derivative of a function is positive on an interval, the function is increasing on that interval. If the first derivative is negative, the function is decreasing. By analyzing the sign of the first derivative, we can determine where the function is increasing or decreasing.

6. To determine the left-hand limit and right-hand limit of a function at a point c, we need to evaluate the function as it approaches c from the left (left-hand limit) and from the right (right-hand limit). The left-hand limit is denoted as lim(x→c⁻) f(x), and the right-hand limit is denoted as lim(x→c⁺) f(x).

7. To find the derivative of a function using the definition of the derivative, we use the formula:

f'(x) = lim(h→0) [(f(x+h) - f(x))/h]

where h is a small increment in x.

8.
a. f'(x) = 2
b. f'(x) = 2x
c. f'(x) = 3x^2
d. f'(x) = 1 (for x > -2)

9.
a. f'(x) = 2x cos(x) - (x^2 - 5) sin(x)
b. g'(x) = (1/2) x^(-1/2) sec(x) + x^(-1/2) sec(x) tan(x)
c. f'(x) = (6x^2 - 10) / (x^2 + 3)^2
d. g'(x) = 2 / tan(x) - 2x / (tan(x))^2

10.
a. The graph of f(x) = (x^2 - 1)^2 is a parabola opening upwards with a vertex at (0, 1).
b. The graph of g(x) = e^x / x is a curve that approaches 0 as x approaches negative infinity and increases without bound as x approaches positive infinity.
c. The graph of g(x) = (x^3 - 3x^2 + 4) / (x^2 - 1) is a rational function with vertical asymptotes at x = 1 and x = -1.