Exercises for Question Type 4: Finding parameter values so that a limit exists (using L’Hôpital) - IB | RevisionDojo

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Question Type 4: Finding parameter values so that a limit exists (using L’Hôpital)

Question Type 4: Finding parameter values so that a limit exists (using L’Hôpital) Exercises

Question 1

Determine in terms of $k$ :

\lim_{x\to0}\frac{\tan(kx) - kx}{x^3}

and state for which $k$ it exists.

[4]

Question 2

Find $k$ such that the limit exists and determine its value:

\lim_{x\to0}\frac{kx - \ln(1 + x)}{x^2}

[5]

Question 3

Determine

\lim_{x\to1}\frac{x^k - 1}{x - 1}

and state for which $k$ this limit exists.

[4]

Question 4

Find all $k$ such that the limit exists:

\lim_{x\to0}\frac{x - \sin(kx)}{x^3}

[5]

Question 5

Determine $k$ so that the limit is finite and compute it:

\lim_{x\to\infty}\frac{kx^3 + x}{x^2 + kx}

[4]

Question 6

Find the values of $k$ such that the limit exists:

\lim_{x\to 0}\frac{k^2x - kx}{4x^2}.

[4]

Question 7

For which integer values of $k$ does the following limit exist, and what is its value?

\lim_{t\to\frac{\pi}{2}}\frac{\cos(kt)}{t-\frac{\pi}{2}}

[5]

Question 8

Find the value of $k$ such that the following limit exists, and compute the limit:

$\lim_{x\to0}\frac{k x^2 + x\sin x - x^2}{x^3}$

[4]

Question 9

Find $k$ such that the limit exists and calculate it:

\lim_{x\to0}\frac{(x - \sin x)-k x^3}{x^5}.

[6]

Question 10

Let $a$ be a constant. Find $b$ so that the limit exists and compute it:

\lim_{x\to0}\frac{a x^3 + b x^2}{x - \sin x}.

[5]