Determine the following limits.​

lim h→0 (h + 6)^2 + (h + 6) − 42 / h

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\left\lbrack1,4\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\left\lbrack1,2\right\rbrack$

Determine the following limits.​

lim x→a (3x + 1)^2 − (3a + 1)^2 / x − a, where a is constant

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\left\lbrack0,3\right\rbrack$

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

Make a conjecture about the values of ${\displaystyle\lim_{x\to2^{-}}g\left(x\right)}$, ${\displaystyle\lim_{x\to2^{+}}g\left(x\right)}$, and ${\displaystyle\lim_{x\to2}g\left(x\right)}$ or state that they do not exist.

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\left\lbrack0,h\right\rbrack$, where $h\gt{0}$ is a real number

Suppose g(x) = {2x+1 if x≠0

5 if x=0.

Compute g(0) and lim x→0 g(x)

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{-}}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{+}}f\left(x\right)$

Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0≤t≤60. <FIGURE>

Suppose p and q are polynomials. If lim x→0 p(x) / q(x)=10 and q(0)=2, find p(0).

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

a. f(1)

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

d. ${\lim }_{x\to 1}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

h. ${\lim }_{x\to 3}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

l. ${\lim }_{x\to 2}f\left(x\right)$

Determine the following limits. 

lim t→∞ (5t² + t sin t) / t²

Use the graph of $g\left(x\right)$ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

$\lim_{x\to2}g\left(x\right)$

Use the graph of g(x) in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 4}g\left(x\right)$

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

lim x→4(3x−7)

Determine the following limits. 

lim x→−∞ (5 + 100/x + sin⁴ x³ / x²)

Determine the following limits.​

lim x→3 x^4 − 81 / x − 3

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

lim x→−95x

Determine the following limits.

a. lim x→2^+ 1 x − 2

Determine the following limits.​

lim p→1 p^5 − 1 / p − 1

Determine the following limits.

b. lim x→3^− 2/(x − 3)^3

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

lim x→1(2x^3−3x^2+4x+5)

Determine the following limits.

a. lim x→4^+ x − 5 / (x − 4)^2

Determine the following limits.

b. lim x→4^− x − 5 / (x − 4)^2

Determine the following limits.

c. lim x→4 x − 5 / (x − 4)^2