Before the midterm

Comparing the wrong leading terms at infinity

The habit: For large x the fastest-growing piece of each part decides; keep it, and in a quotient divide the top and the bottom by it.

19 practice questions are built to catch it, in 4 topics.

Practise against it: 6 questions

What it looks like

A question from power functions and first sketches, with the answer this slip gives beside the right one.

Introductory1 mark · 2 min

Let $f(x) = 3x^{2} - x^{5}$ . For very large positive $x$ , which term dominates, and what is the sign of $f(x)$ ?

With the slip: The 3x² term dominates, so f(x) is positive.
Worked through: The x⁵ term dominates, so f(x) is negative.

The coefficient 3 only scales x²: at x = 10, 3x² = 300 while x⁵ = 100 000. The higher power wins for large x, and the sign it carries in f decides the sign of f(x).

See the worked solution

Where it comes up

Power functions and first sketches9 questions
Limits5 questions
L’Hôpital’s rule2 questions
Curve sketching3 questions

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Let $g(x) = x^{4} + 6x$ . Close to $x = 0$ , which term dominates, and what does the graph look like there?
Power functions and first sketchesIntroductory
02
Let $f(x) = x^{3} + 4\sin x$ , with $x$ in radians. Which term dominates close to $x = 0$ , and which dominates for large $|x|$ ?
Power functions and first sketchesIntroductory
03
The graph of $y = ax^{2} + bx^{4}$ is shown, where $a$ and $b$ are nonzero constants. What are the signs of $a$ and $b$ ?
Power functions and first sketchesIntroductory
04
The speed of a reaction depends on the concentration $x$ of a chemical according to the function below. As $x$ grows large, $y$ approaches a limiting value. Find the concentration $x > 0$ at which $y$ is exactly half of that limiting value. $\displaystyle y = \frac{6x^{3}}{2 + x^{3}}$
Power functions and first sketchesStandard
05
For all sufficiently large $x$ , which ordering of the three functions $e^{x}$ , $x^{100}$ and $\log x$ is correct, from largest to smallest?
Power functions and first sketchesStandard
06
Each of $(\log x)^{4}$ , $\sqrt{x}$ and $1.005^{x}$ grows without bound as $x \to \infty$ . For all sufficiently large $x$ , which ordering is correct, from largest to smallest?
Power functions and first sketchesStandard

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