Question

In Exercises 29 and 30 , use a graph to show that the limit does not exist. $$\lim _ { x \rightarrow 2 } \frac { x + 1 } { x ^ { 2 } - 4 }$$

Short answer

The limit \( \lim _ { x \rightarrow 2 } \frac { x + 1 } { x ^ { 2 } - 4 }\) does not exist. This is because if we observe the graph as \(x\) approaches \(2\), from either side, the function does not converge to a single value. The function is not defined at \(x = 2\) itself as that makes the denominator zero, giving the function an undefined value.

Step-by-step solution

Identify the Function

Identify the function that is to be graphed. In this case it is \(f(x) = \frac{x+1}{x^2-4}\)

Identify Points of Discontinuities

Discontinuity occurs when denominator of the function is equal to zero. Here, find the values of \(x\) which make \(x^2-4 = 0\). Solve this equation to find that \(x = +2, -2\) are points of discontinuity.

Graph the Function

Graph the function \(f(x) = \frac{x+1}{x^2-4}\). Plot points around \(x = -2, 2\) (include points on left and right side of these points) to see how function value changes as we approach the points of discontinuity.

Observing the Graph

Observe the graph carefully around \(x=2\). If the function approaches different values from left and right side of \(x = 2\), or makes a jump at that point or goes to infinity, then the limit does not exist.