Use limit notation to describe the following sentence:
The value that the function y = x + 4 approaches as x approaches 168.
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) |
Hence, 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. Give your answers correct to four decimal places.
| x | -0.1 | -0.01 | -0.001 | 0.001 | 0.01 | 0.1 |
|---|---|---|---|---|---|---|
| f(x) |
Hence, 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(x) |
Hence, find the value of \lim_{x \to 4}\left(\dfrac{x^{2} - 4 x}{x - 4}\right).
Consider \lim_{x \to - 5 }\left(\dfrac{x^{2} + 4}{x + 5}\right).
Complete the table to find the values of f \left( x \right) as x gets closer and closer to 5 from the left, and closer and closer to 5 from the right:
| x | -5.1 | -5.01 | -5.001 | -4.999 | -4.99 | -4.9 |
|---|---|---|---|---|---|---|
| \dfrac{x^2+4}{x+5} |
Does the above limit exist?
The graph of the function \dfrac{x^{2} - 1}{x + 1} is shown below:
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?
Hence, write a sentence to describe the limiting behaviour of \dfrac{x^{2} - 1}{x + 1} at x=-1.
Write this same sentence as an equation, using limit notation.
Consider the graph of a function, f \left( x \right):
Does \lim_{x \to 3} f \left( x \right) exist? If yes, state the value of the limit.
Consider the graph of a function, f \left( x \right):
Does \lim_{x \to 2} f \left( x \right) exist? If yes, state the value of the limit.
Consider the graph of a function, f \left( x \right):
Does \lim_{x \to 3} f \left( x \right) exist? If yes, state the value of the limit.
Evaluate:
\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 5} \sqrt{41 - x}
\lim_{x \to 1}\left( 3 x^{4} - 5 x^{3} + 2\right)
\lim_{x \to - 4 }\left(\dfrac{x^{2} + 6 x + 8}{x + 4}\right)
\lim_{x \to 3}\left(\dfrac{x^{2} - 9 x + 18}{x - 3}\right)
\lim_{x \to 0}\left(\dfrac{x^{2} - 4 x}{x}\right)
\lim_{x \to - 3 }\left(\dfrac{5 x + 3}{3 x + 4}\right)
\lim_{x \to - 5 }\left(\dfrac{x^{2} - 25}{x + 5}\right)
\lim_{x \to - 3 }\left(\dfrac{x^{2} + 3 x}{x^{2} - 9}\right)
\lim_{x \to 3}\left(\dfrac{x^{2} - 5 x + 6}{x - 3}\right)
\lim_{x \to - 5 }\left(\dfrac{x^{2} + x - 20}{x^{2} - 25}\right)
\lim_{x \to - 5 }\left(\dfrac{\left(x + 5\right)^{2}}{x^{2} + 2 x - 15}\right)
\lim_{x \to 0}\left(\dfrac{x^{2} - 36}{x^{2} + 36}\right)
\lim_{x \to 3}\left(\dfrac{x^{3} - 27}{x^{4} - 81}\right)
\lim_{x \to 4}\left(\dfrac{x - 4}{\sqrt{x} - 2}\right)
\lim_{x \to 5}\left(\dfrac{\dfrac{1}{x} - \dfrac{1}{5}}{x - 5}\right)
\lim_{x \to - 9 }\left(\dfrac{x^{3} + 18 x^{2} + 81 x}{x^{2} + 5 x - 36}\right)
\lim_{x \to 3}\left(\dfrac{\left(x - 3\right)^{2}}{3 x^{2} - 13 x + 12}\right)
Consider the function f \left( x \right) = \dfrac{x^{2} + 7 x + 10}{x + 5}.
Evaluate \dfrac{x^{2} + 7 x + 10}{x + 5} at x = - 5.
Evaluate \lim_{x \to - 5 }\left(\dfrac{x^{2} + 7 x + 10}{x + 5}\right).