Question

Use the given graph to determine the limit, if it exists. A coordinate graph is shown with a horizontal line crossing the y axis at five that ends at the open point 2, 5, a closed point at 2, 1, and another horizontal line starting at the open point 2, negative 2 and continues to the right. Find limit as x approaches two from the left of f of x. and limit as x approaches two from the right of f of x..

Answer 1

`\lim_(x \to 2^-)f(x)=5`

`\lim_(x \to 2^+)f(x)=-2`

Explanation:

The graph of f(x) is that of a piecewise function that is composed of 2 horizontal lines and a point.

Notice in the graph that:

`f(x) = 5` if
`x <2`

`f(x) = 1` if
`x = 2`

`f(x) = -2` if
`x> 2`

We must find the limit on the left of 2 and on the right of 2.

When we find the limit on the left of 2 it means that x is a value infinitesimally smaller than 2. Then
`x <2` .

Since x is less than 2 then f(x) tends to 5.

So

`\lim_(x \to 2^-)f(x)\\\\=\lim_(x \to 2^-)5 = 5`

When we find the limit on the right of 2, it means that x is a value infinitesimally larger than 2. Then
`x> 2`.

Since x is greater than 2 then f(x) tends to -2.

So

`\lim_(x \to 2^+)f(x)\\\\=\lim_(x \to 2^+)-2 = -2`

Answer 2

The limit as x approaches 2 from the left of f(x) is 5

The limit as x approaches 2 from the right of f(x) is -2

Explanation:

From the left, the coordinate graph shows a horizontal line crossing the y axis at five that ends at the open poing (2,5).

The limit as x approaches 2 from the left of f(x) is 5. Given that, from the left the graph approaches to 5.

From the right, the coordinate graph shoes a horizontal line starting at the open point (2, -2) and continues to the right.

The limit as x approaches 2 from the right of f(x) is -2. Given that, from the right the graph approaches to 5.