Consider the function f \left( x \right) = \dfrac{1}{7 - x}.
Complete the first table of values, in which x \lt 7.
Complete the second table of values, in which x \gt 7.
Find the limit of f \left( x \right) as the value of x approaches 7.
| x | 5 | 6 | 6.9 | 6.99 |
|---|---|---|---|---|
| f(x) |
| x | 9 | 8 | 7.1 | 7.01 |
|---|---|---|---|---|
| f(x) |
Consider the function f \left( x \right) = 5 x^{2} + 1:
Complete the table to find the exact values of f \left( x \right) as x gets closer and closer to 2 from the left, and closer and closer to 2 from the right:
| x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
|---|---|---|---|---|---|---|
| f(x) |
Find the value of \lim_{x \to 2}\left( 5 x^{2} + 1\right).
Consider the function f \left( x \right) = \dfrac{2 - x}{x^{2} + 2}:
Complete the table to find the values of f \left( x \right) as x gets closer and closer to 0 from the left, and closer and closer to 0 from the right. Round your answers to four decimal places.
| x | -0.1 | -0.01 | -0.001 | 0.001 | 0.01 | 0.1 |
|---|---|---|---|---|---|---|
| f(x) |
Find the value of \lim_{x \to 0}\left(\dfrac{2 - x}{x^{2} + 2}\right).
Consider the function f \left( x \right) = \dfrac{x^{2} - 4 x}{x - 4}.
Complete the table to find the values of f \left( x \right) as x gets closer and closer to 4 from the left, and closer and closer to 4 from the right:
| x | 3.9 | 3.99 | 3.999 | 4.001 | 4.01 | 4.1 |
|---|---|---|---|---|---|---|
| f \left( x \right) |
Find the value of \lim_{x \to 4}\left(\dfrac{x^{2} - 4 x}{x - 4}\right).
Consider the function f \left( x \right) = \dfrac{x^{3} + x + 2}{x + 1}.
Complete the following table:
| x | -1.1 | -1.01 | -1.001 | -0.999 | -0.99 | -0.9 |
|---|---|---|---|---|---|---|
| f \left( x \right) |
Find \lim_{x \to - 1 } f \left( x \right).
Consider the function f \left( x \right) = \dfrac{\sqrt{x} + 4}{x - 5} :
Complete the following table, rounding all values to two decimal places:
| x | 4.9 | 4.99 | 4.999 | 5.001 | 5.01 | 5.1 |
|---|---|---|---|---|---|---|
| f \left( x \right) |
Does \lim_{x \to 5} f \left( x \right) exist? Explain your answer.
Consider the limit: \lim_{x \to - 5 }\left(\dfrac{x^{2} + 4}{x + 5}\right).
Complete the table.
| x | -5.1 | -5.01 | -5.001 | -5 | -4.999 | -4.99 | -4.9 |
|---|---|---|---|---|---|---|---|
| \dfrac{x^2+4}{x+5} | - |
Does the above limit exist? Explain your answer.
Consider the limit: \lim_{x \to 3}\left(\dfrac{e^{x - 3} + x - 4}{x - 3}\right).
Complete the following table:
| x | 2.9 | 2.99 | 2.999 | 3 | 3.001 | 3.01 | 3.1 |
|---|---|---|---|---|---|---|---|
| \left(\dfrac{e^{x - 3} + x - 4}{x - 3}\right) | - |
Does the above limit exist? Explain your answer.
Consider the graph of the piecewise function g \left( x \right):
If we start at - 4 and move along g \left( x \right) to the right towards x = - 2, what y-value do we approach?
If we start at 0 and move along g \left( x \right) to the left towards x = - 2, what y-value do we approach?
Does the \lim_{x \to -2} g(x) exist? Explain your answer.
Consider the graph of the function f\left(x\right) = \dfrac{x^{2} - 1}{x + 1}:
If we start at x = - 3 and move along the function to the right towards x = -1, what y-value do we approach?
If we start at x = 1 and move along the function to the left towards x = -1, what y-value do we approach?
Describe the behaviour of \\ f\left(x\right) = \dfrac{x^{2} - 1}{x + 1} as x approaches -1.
Write part (c) using limit notation.
Consider the graph of f(x):
Does the limit \lim_{x \to 3} f \left( x \right) exist? Explain your answer.
Consider the graph of f(x):
Does the limit \lim_{x \to 0} f \left( x \right) exist? Explain your answer.
Consider the graph of y = \dfrac{x + 3}{\left(x - 5\right)^{2}}:
Does the limit \lim_{x \to 5}\left(\dfrac{x + 3}{\left(x - 5\right)^{2}}\right) exist? Explain your answer.
Consider the graph of y = \dfrac{x^{2}}{1 - \cos x}:
Does the limit \lim_{x \to 0}\left(\dfrac{x^{2}}{1 - \cos x}\right) exist? Explain your answer.
Consider the graph of the function f \left( x \right):
Find the value of \lim_{x \to 3} f \left( x \right).
Consider the graph of the function f \left( x \right):
Find the value of \lim_{x \to 0} f \left( x \right).
Consider the graph of the function f \left( x \right):
Find the value of \lim_{x \to 3} f \left( x \right).
Consider the graph of the function f \left( x \right):
Find the value of \lim_{x \to 3} f \left( x \right).
Consider the graph of the function f \left( x \right):
Find the value of \lim_{x \to 2} f \left( x \right).
Consider the graph of the function \\ f \left( x \right) = \dfrac{1}{x + 4}:
Find the value of:
Consider the function f \left( x \right) = \dfrac{1}{x + 3}:
Find the value of \dfrac{1}{x + 3} at x = - 3.
Find the value of \lim_{x \to - 3 }\left(\dfrac{1}{x + 3}\right).
Consider the function f \left( x \right) = \dfrac{x^{2} + 7 x + 10}{x + 5} :
Find the value of f\left(x\right) at x = - 5.
Find the value of \lim_{x \to - 5 } f\left(x\right).
Find the value of the following limits:
\lim_{x \to 5} 9
\lim_{x \to 3}\left( - 2 \right)
\lim_{x \to 6}\left( 5 x\right)
\lim_{x \to 6}\left( 4 x^{2}\right)
\lim_{x \to 0}\left(\dfrac{x^{2} - 6 x}{x}\right)
\lim_{x \to 1}\left( 3 x^{4} - 5 x^{3} + 2\right)
\lim_{x \to 5} \sqrt{41 - x}
\lim_{x \to - 3 }\left(\dfrac{2 x + 5}{5 x + 2}\right)
\lim_{x \to 0}\left(\dfrac{x^{2} - 25}{x^{2} + 25}\right)
\lim_{x \to 3}\left(\dfrac{x^{2} - 9 x + 18}{x - 3}\right)
\lim_{x \to - 4 }\left(\dfrac{x^{2} + 6 x + 8}{x + 4}\right)
\lim_{x \to - 2 }\left(\dfrac{x^{2} + 5 x + 6}{x + 2}\right)
\lim_{x \to 3}\left(\dfrac{x^{2} - x - 6}{x^{2} - 9}\right)
\lim_{x \to - 2 }\left(\dfrac{x^{2} - 4}{x + 2}\right)
\lim_{x \to - 9 }\left(\dfrac{x^{2} + 9 x}{x^{2} - 81}\right)
\lim_{x \to 3}\left(\dfrac{\left(x - 3\right)^{2}}{x^{2} - 8 x + 15}\right)
\lim_{x \to - 4 }\left(\dfrac{\left(x + 4\right)^{2}}{5 x^{2} + 24 x + 16}\right)
Consider the function f \left( x \right) = \dfrac{3}{x + 2}.
What value does the function approach as x approaches infinity?
What value does the function approach as x approaches negative infinity?
What value does the function approach as x approaches zero?
Consider the function f \left( x \right) = 3^{x} :
Find the limit of the function as x approaches infinity.
Find the limit of the function as x approaches negative infinity.
Find the limit of the function as x approaches zero.
Consider the graph of the function:
What value does y approach as x approaches infinity?
What value does y approach as x approaches negative infinity?
What value does y approach as x approaches zero?
Consider the graph of the function:
Find the limit of y as x approaches infinity.
Find the limit of y as x approaches negative infinity.
Find the limit of y as x approaches zero.
Use limit notation to represent the following statement:
The value that the function y = x + 4 approaches as x approaches 168
The price P \left( t \right) of coffee in a cafe changes as a function of time, where t is in years.
If P(t)=4 \times 1.025^t, find the amount that the price of coffee approaches after 13 years.
A ball is dropped, and on the first bounce it reaches a height of 2 \text{ m} and after each bounce it reaches one quarter of its previous height. On the nth bounce, it travels a distance of d \left( n \right).
Write an expression for d(n) and find the value that the distance travelled by the ball on each bounce approaches as the number of bounces n goes on forever.