Questions: In Problems 39-42, graph each function with a graphing calculator and use it to predict the limit. Check your work either by using the table feature of the calculator or by finding the limit algebraically. 39. lim x→10 (x^2-19x+90)/(3x^2-30x) 40. lim x→-3 (x^4+3x^3)/(2x^4-18x^2)

Solution Steps

The given limit is: \[ \lim _{x \rightarrow 10} \frac{x^{2}-19 x+90}{3 x^{2}-30 x} \]

First, factor both the numerator and the denominator.

The numerator \(x^2 - 19x + 90\) can be factored as \((x - 9)(x - 10)\).

The denominator \(3x^2 - 30x\) can be factored as \(3x(x - 10)\).

After factoring, the expression becomes: \[ \frac{(x - 9)(x - 10)}{3x(x - 10)} \]

Cancel the common factor \((x - 10)\): \[ \frac{x - 9}{3x} \]

Now, evaluate the limit as \(x\) approaches 10: \[ \lim _{x \rightarrow 10} \frac{x - 9}{3x} = \frac{10 - 9}{3 \times 10} = \frac{1}{30} \]

Final Answer