Before the midterm

Taking √x² = x for negative x

The habit: The square root of x² is |x|, which is minus x when x is negative.

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

Practise against it: 6 questions

What it looks like

A question from limits, with the answer this slip gives beside the right one.

Standard2 marks · 4 min

Find the horizontal asymptote of the graph of the function below as $x \to -\infty$ : enter the value of $y$ it approaches.

f(x) = \frac{\sqrt{9x^{2} + 4}}{2x - 1}

With the slip: $\displaystyle 1.5$
Worked through: $\displaystyle y = -\frac{3}{2}$

3/2 is the level as x → ∞. For x < 0, √(9x²) = 3|x| = −3x: compare the signs of the top and of 2x − 1 there.

See the worked solution

Where it comes up

Limits5 questions
Continuity1 question
The definition of the derivative1 question
Curve sketching1 question

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
On which set is $g(x) = \dfrac{1}{\sqrt{x^{2} - 4}}$ continuous?
ContinuityIntroductory
02
Find the horizontal asymptote of the graph of the function below as $x \to -\infty$ : enter the value of $y$ it approaches. $\displaystyle f(x) = \frac{4x + 1}{\sqrt{9x^{2} + 2}}$
LimitsStandard
03
Let $f(x) = x\,|x|$ . Which statement about $f$ at $x = 0$ is correct?
The definition of the derivativeStandard
04
Evaluate the limit. $\displaystyle \lim_{x \to -\infty} \frac{2x + 3}{\sqrt{9x^{2} + 1} - x}$
LimitsChallenging
05
Evaluate the limit. $\displaystyle \lim_{x \to -\infty} \Bigl(\sqrt{x^{2} + 5x} - \sqrt{x^{2} - 3x}\Bigr)$
LimitsChallenging
06
For exactly one pair of constants $a$ and $b$ , the limit below exists and equals 2. Find $a$ and $b$ , and enter $a + b$ . $\displaystyle \lim_{x \to -\infty} \Bigl(\sqrt{36x^{2} + ax + 5} + bx\Bigr) = 2$
LimitsCapstone