Question

In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { t \rightarrow 2 } \frac { t ^ { 2 } - 3 t + 2 } { t ^ { 2 } - 4 }$$

Short answer

The limit of the function as t approaches 2 is \( \frac {1} {4} \).

Step-by-step solution

Graphical Interpretation

A graph of the function \(f(t) = \frac { t ^ { 2 } - 3 t + 2 } { t ^ { 2 } - 4 }\) is sketched. You examine the behavior of the function as it approaches \(t = 2\). The y-coordinate of the point on the graph where \(t = 2\) is the graphical interpretation of the limit.

Factorize the function

Factorizing the numerator and the denominator of the function, \(f(t)= \frac { (t-1)(t-2) } { (t+2)(t-2) }\)

Cancelling common factors

Simplify the equation by cancelling out the common factors, in this case \(t-2\). Therefore, \(f(t)= \frac {t-1} {t+2}\).

Substitute the value of t

After simplifying the function, substitute the value \(t=2\) into the function. \(f(t)= \frac {2-1} {2+2}\), which simplifies to \(f(t)= \frac {1} {4}\).