Basic Limits & Continuity (DP IB Analysis & Approaches (AA): HL): Exam Questions

For each of the following, either show that the limit converges and find its value, or else explain why the limit diverges:

$\underset{x\to \frac{2}{5}}{\lim }$$\frac{1}{4x^2-25}$

$\underset{x\to \frac{5}{2}}{\lim }$$\frac{1}{4x^2-25}$

$\underset{x\to \frac{5}{2}}{\lim }$$\frac{2x-5}{4x^2-25}$

Evaluate the limit

 $\underset{x\to -\infty }{\lim }\left(\frac{5+{\left(2x-3\right)}^2}{2x^2}\right)$ 

justifying your answer by clear mathematical reasoning.

(i) Show that the limit

 $\underset{x\to +\infty }{\lim }\frac{x^2-x+3}{x}$

   diverges.  Be sure to show clear algebraic working.

(ii) Determine any asymptotes on the graph of the curve with equation          

 $y=\frac{x^2-x+3}{x}$

A student has attempted to evaluate the limit

$\underset{x\to +\infty }{\lim }\left(\frac{x^2+x}{x^3}\right)$

as follows:

$\underset{x\to +\infty }{\lim }\left(\frac{x^2+x}{x^3}\right)=\frac{{\left(+\infty \right)}^2+\left(+\infty \right)}{{\left(+\infty \right)}^3}=\frac{+\infty }{+\infty }=1$

Explain what is wrong with the student’s work.

Determine the correct evaluation of the limit, justifying your answer by clear mathematical reasoning.

Use technology to help you sketch the graph of $y=\frac{x^2+x}{x^3}$, and show that the graph confirms your answer to part (b).

Consider the function $f$ defined by

$f\left(x\right)=\frac{1}{{\left(2x+6\right)}^2}$ 

Evaluate the limits

(i) $\underset{x\to -3^{-}}{\lim }f\left(x\right)$

(ii) $\underset{x\to -3^{+}}{\lim }f\left(x\right)$

Evaluate the limits

(i) $\underset{x\to -\infty }{\lim }f\left(x\right)$

(ii) $\underset{x\to +\infty }{\lim }f\left(x\right)$

Use your results from parts (a) and (b) to write down the equations of any asymptotes on the graph of $y=f\left(x\right)$.

Use technology to help you sketch the graph of $y=f\left(x\right)$,  and show that this confirms your results from parts (a), (b) and (c).

Consider the function $g$ defined by

 $g\left(x\right)=\frac{1}{x^3-8}+1$

Evaluate the limits

(i) $\underset{x\to 2^{-}}{\lim }g\left(x\right)$

(ii) $\underset{x\to 2^{+}}{\lim }g\left(x\right)$

Evaluate the limits

(i) $\underset{x\to -\infty }{\lim }g\left(x\right)$

(ii) $\underset{x\to +\infty }{\lim }g\left(x\right)$

Use your results from parts (a) and (b) to write down the equations of any asymptotes on the graph of $y=g\left(x\right)$.

Use technology to help you sketch the graph of $y=g\left(x\right)$,  and show that this confirms your results from parts (a), (b) and (c).

The function $f$ is a piecewise function defined by 

 $f\left(x\right)=\{\begin{array}{ll}3x-7,& x<3\\ 1,& x=3\\ x^2-8x+17,& x>3\end{array}$

Explain why $f$ is not continuous at $x=3$.

Give an example of a function $g$ that is continuous for all values of $x\in \mathrm{ℝ}$, but is not differentiable for all values of $x\in \mathrm{ℝ}$.  Include a sketch of the graph of the function, identifying the point(s) where the function is not differentiable.

For each of the following, either show that the limit converges and find its value, or else explain why the limit diverges:

$\underset{x\to 0}{\lim }\tan \left(x-\frac{\mathrm{\pi }}{4}\right)$

$\underset{x\to \frac{3\mathrm{\pi }}{4}}{\lim }\tan \left(x-\frac{\mathrm{\pi }}{4}\right)$

$\underset{x\to \frac{3\mathrm{\pi }}{4}}{\lim }\frac{\tan \left(x-{\displaystyle \frac{\pi }{4}}\right)}{\sec \left(x-{\displaystyle \frac{\pi }{4}}\right)}$

Evaluate the limit

 $\underset{x\to +\infty }{\lim }\cos \left(\frac{3}{x^2}\right)$

justifying your answer by clear mathematical reasoning.

(i) Show that the limit

 $\underset{x\to -\infty }{\lim }\left(\frac{2x^3+6x^2+1}{2x^2}+\tan \left(\frac{\pi x^3-2x^2+3}{7-2x-4x^3}\right)\right)$ 

diverges.  Be sure to show clear algebraic working.

(ii) Determine the asymptotic behaviour of the curve with equation

 $y=\frac{2x^3+6x^2+1}{2x^2}+\tan \left(\frac{\pi x^3-2x^2+3}{7-2x-4x^3}\right)$        

as $x\to \pm \infty$.

A student has attempted to evaluate the limit

 $\underset{x\to -\infty }{\lim }\left(\frac{x^2+x+14}{x^2+x-2}\right)$ 

as follows:

$\underset{x\to -\infty }{\lim }\left(\frac{x^2+x+14}{x^2+x-2}\right)=\frac{{\left(-\infty \right)}^2+\left(-\infty \right)+14}{{\left(-\infty \right)}^2+\left(-\infty \right)-2}=\frac{\left(+\infty \right)+\left(-\infty \right)+14}{\left(+\infty \right)+\left(-\infty \right)-2}=\frac{0+14}{0-2}=-7$

Explain what is wrong with the student’s work.

Determine the correct evaluation of the limit, justifying your answer by clear mathematical reasoning.

Use technology to help you sketch the graph of $y=\frac{x^2-x+14}{x^2-x-2}$, and show that the graph confirms your answer to part (b).

Consider the function $f$ defined by

$f\left(x\right)=\frac{1}{{\left(\arctan x\right)}^2}$ 

Evaluate the limits

(i) $\underset{x\to 0^{-}}{\lim }f\left(x\right)$

(ii) $\underset{x\to 0^{+}}{\lim }f\left(x\right)$

Evaluate the limits

(i) $\underset{x\to -\infty }{\lim }f\left(x\right)$

(ii) $\underset{x\to +\infty }{\lim }f\left(x\right)$

Use your results from parts (a) and (b) to write down the equations of any asymptotes on the graph of $y=f\left(x\right)$.

Use technology to help you sketch the graph of $y=f\left(x\right)$,  and show that this confirms your results from parts (a), (b) and (c).

Consider the function g defined by

 $g\left(x\right)=\frac{2x-3}{x}-\frac{1}{x^3+1}$

Evaluate the limits

(i) $\underset{x\to -1^{-}}{\lim }g\left(x\right)$

(ii) $\underset{x\to -1^{+}}{\lim }g\left(x\right)$

Evaluate the limits

(i) $\underset{x\to -\infty }{\lim }g\left(x\right)$

(ii) $\underset{x\to +\infty }{\lim }g\left(x\right)$

Write down the equations of any asymptotes on the graph of $y=g\left(x\right)$.

Use technology to help you sketch the graph of $y=g\left(x\right)$,  and show that this confirms your results from parts (a), (b) and (c).

The function $f$ is a piecewise function defined by

$f\left(x\right)=\{\begin{array}{ll}10-3x,& x<2\\ \frac{x^2-4}{x-2},& x=2\\ \left|x^2-2x-4\right|,& x>2\end{array}$

 Explain why $f$ is not continuous at $x=2$.

Give an example of a function $g$ that is continuous for all $x\in \mathrm{ℝ}$,  but which is not differentiable at  $x=3$.  Include a sketch of the graph of the function, identifying all points where the function is not differentiable.

Write down a continuous function $h$ for which $\underset{x\to -\infty }{\lim }h\left(x\right)$  and $\underset{x\to +\infty }{\lim }h\left(x\right)$  both exist and are finite, but for which $\underset{x\to -\infty }{\lim }h\left(x\right)\ne \underset{x\to +\infty }{\lim }h\left(x\right)$.