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Date	November 2018	Marks available	2	Reference code	18N.2.SL.TZ0.S_3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Hence or otherwise and Write down	Question number	S_3	Adapted from	N/A

Question

Let $f (x) = \frac{6 x - 1}{2 x + 3}$ $f\left( x \right) = \frac{{6x - 1}}{{2x + 3}}$ , for $x \neq - \frac{3}{2}$ $x \ne - \frac{3}{2}$ .

For the graph of $f$ $f$ , find the $y$ $y$ -intercept.

[2]

a.i.

Hence or otherwise, write down $lim x \to \infty (\frac{6 x - 1}{2 x + 3})$ $\mathop {{\text{lim}}}\limits_{x \to \infty } \left( {\frac{{6x - 1}}{{2x + 3}}} \right)$ .

[2]

Markscheme

valid method (M1)

eg $f$ $f$ (0), sketch of graph

$y$ $y$ -intercept is $- \frac{1}{3}$ $- \frac{1}{3}$ (exact), −0.333, $(0, - \frac{1}{3})$ $\left( {0,\, - \frac{1}{3}} \right)$ A1 N2

[2 marks]

a.i.

valid approach (M1)

eg recognizing that $lim x \to \infty f (x)$ $\mathop {{\text{lim}}}\limits_{x \to \infty } f\left( x \right)$ is related to the horizontal asymptote,

table with large values of $x$ $x$ , their $y$ $y$ value from (a)(iii), L’Hopital’s rule $lim x \to \infty f (x) = 3$ $\mathop {{\text{lim}}}\limits_{x \to \infty } f\left( x \right) = 3$ .

$lim x \to \infty (\frac{6 x - 1}{2 x + 3}) = 3$ $\mathop {{\text{lim}}}\limits_{x \to \infty } \left( {\frac{{6x - 1}}{{2x + 3}}} \right) = 3$ A1 N2

[2 marks]

Examiners report

[N/A]

a.i.

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.8—Reciprocal and simple rational functions, equations of asymptotes

Show 26 related questions

18N.2.AHL.TZ0.H_8a:
On the following axes, sketch the two possible graphs of $y = f (x)$ $y = f\left( x \right)$ giving the equations of any asymptotes in terms of $p$ $p$ and $q$ $q$ .
22M.1.SL.TZ2.4a.ii:
Write down the equation of the horizontal asymptote.
22M.1.SL.TZ2.4a.i:
Write down the equation of the vertical asymptote.
22M.1.SL.TZ2.4b:
On the set of axes below, sketch the graph of $y = f (x)$ $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
22M.1.AHL.TZ2.3a.i:
Write down the equation of the vertical asymptote.
21N.1.SL.TZ0.3c:
Sketch the graph of $f$ $f$ on the axes below.
19M.1.AHL.TZ2.H_5a:
Sketch the graph of $y = \frac{x - 4}{2 x - 5}$ $y = \frac{{x - 4}}{{2x - 5}}$ , stating the equations of any asymptotes and the coordinates of any points of intersection with the axes.
21N.1.AHL.TZ0.2a.i:
the vertical asymptote of the graph of $f$ $f$ .
21N.1.AHL.TZ0.2a.ii:
the horizontal asymptote of the graph of $f$ $f$ .
21N.1.AHL.TZ0.2c:
Sketch the graph of $f$ $f$ on the axes below.
21N.1.SL.TZ0.3a.i:
the vertical asymptote of the graph of $f$ $f$ .
21N.1.SL.TZ0.3a.ii:
the horizontal asymptote of the graph of $f$ $f$ .
21N.1.SL.TZ0.3b.i:
the $x$ $x$ -axis.
21N.1.AHL.TZ0.2b.ii:
the $y$ $y$ -axis.
18N.2.SL.TZ0.S_3a.i:
For the graph of $f$ $f$ , find the $y$ $y$ -intercept.
22M.1.SL.TZ2.4c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.1.AHL.TZ2.3a.ii:
Write down the equation of the horizontal asymptote.
22M.1.AHL.TZ2.3c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.2.SL.TZ1.8a.ii:
find the equation of the horizontal asymptote.
18M.2.SL.TZ2.S_7b:
Write down the equation of the horizontal asymptote to the graph of f.
18M.1.AHL.TZ2.H_10b.i:
Express $g (x)$ $g\left( x \right)$ in the form $A + \frac{B}{x - 2}$ $A + \frac{B}{{x - 2}}$ where A, B are constants.
21N.1.AHL.TZ0.2d:
The function $g$ $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .
22M.2.SL.TZ1.8a.i:
write down the equation of the vertical asymptote.
21N.1.SL.TZ0.3b.ii:
the $y$ -axis.
21N.1.AHL.TZ0.2b.i:
the $x$ -axis.
18N.2.AHL.TZ0.H_8b:
Given that $p = \frac{4}{3}$ , $q = \frac{4}{9}$ and A has coordinates $\left( { - \frac{1}{2},\,\,0} \right)$ , determine the possible sets of values for $a$ , $b$ and $c$ .

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