Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x² − 4x + 3) / (x − 1)

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (x² − 4x + 3) / (x − 1)

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.​​

If limx→a f(x) = L, then f(a)=L.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (2x³ + 10x² + 12x) / (x³ + 2x²)

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.​​

The limit lim x→a f(x) / g(x) does not exist if g(a)=0.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4) 

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

Let​​ f(x) = {x^2+1 / if x<−1

√x+1 if x≥−1.

Compute the following limits or state that they do not exist.

limx→−1 f(x)

Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = x²(4x² − √(16x⁴ + 1))

Evaluate lim x→2^+ √x−2.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x⁴ − 1)/(x^2−1)

Explain why lim x→3^+ √ x−3 / 2−x does not exist.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=√x^2+2x+6−3 / x−1

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=√x^2+2x+6−3 / x−1

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=|1−x^2| / x(x+1)

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=|1−x^2| / x(x+1)

A right circular cylinder with a height of 10 cm and a surface area of S cm² has a radius given by​​ r(S)=1/2(√100+2S/π −10).

Find lim S→0^+ r(S) and interpret your result.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=3e^x+10 / e^x

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=3e^x+10 / e^x

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−1

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→−∞ cot^−1x

a. Use the Intermediate Value Theorem to show that the equation has a solution in the given interval.

x=cos x; (0,π/2)

A sine limit It can be shown that 1−x^2/ 6 ≤ sin x/ x ≤1, for x near 0​​.

Use these inequalities to evaluate lim x→0 sin x/ x.

Use an appropriate limit definition to prove the following limits.

lim x→1 (5x−2) =3;

Suppose f(x) = {x^2 − 5x + 6 / x − 3 if x≠3

a if x=3.

Determine a value of the constant a for which lim x→3 f(x) = f(3).

Suppose ​​ g(x) = {x^2−5x if x≤−1

ax^3−7 if x>−1.

Determine a value of the constant a for which lim x→−1 g(x) exists and state the value of the limit, if possible.

Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.

lim x→1 x^6 − 1 / x − 1