CPM Homework Help : APCALC Problem 3-189

Without a calculator, write the equation of the line tangent to $g ( z ) = \large\frac { z ^ { 7 } + 5 z ^ { 6 } - z ^ { 3 } } { z ^ { 2 } }$ at $z = 0$ .

Compute without a calculator

Hint:

Write the tangent line in point-slope form: $y − y_1 = m(x − x_1)$

Step 1:

Find the point by evaluating $g(0)$ . Careful! Notice the denominator.

Answer:

Though the denominator 'cancels out', there remains a hole in the graph at $z = 0$ . Since there is no point, there is no line, and consequently no tangent line.

Step 2:

Step 2 would be to use $g ^\prime (z)$ to find the slope at $z = 0$ . However, since the tangent line does not exist, there is no reason to find its slope!