Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{1-\cos (2x-2)}{{(x-1)}^2};a=1$

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure.​​ <IMAGE>

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <IMAGE>

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>4

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2+x−2 / x−1; a=1

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

Determine the following limits. 

lim x→∞ (3x¹² − 9x⁷)

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.​ 

a. s(t)=−16t^2+80t+60 at t=3

Determine the following limits. 

lim x→−∞ (3x⁷ + x²) 

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.​ 

c. s(t)=40 sin 2t at t=0

Determine the following limits.​

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Determine the following limits.​

lim x→π/2 1/√sin x − 1 / x + π/2

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim p→2 3p / √4p + 1 − 1

Determine the following limits. 

lim x→∞ x^4+7 / x^5+x^2−x

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 −5x / √4x − 3

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to \infty }{\lim }x\cos \left(x\right)$

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to 0}{\lim }\frac{1-\cos \left(x\right)}{{\cos }^2\left(x\right)-3\cos \left(x\right)+2}$

Determine the following limits.​

lim u→0^+ u − 1 / sin u

Evaluate each limit and justify your answer. 

lim x→0 (x^8−3x^6−1)^40

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

a. Graph the position function, for 0≤t≤9.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground? 

Determine the following limits.​

lim x→0^− 2 / tan x

Evaluate each limit and justify your answer. 

lim t→4 t−4 /√t−2

Determine the following limits. 

lim x→∞ sin x / e^x