IB Mathematics Analysis and Approaches Question Type 4: Finding Parameter Values So That a Limit Exists Exercises

Determine in terms of $k$:

$$\underset{x\to 0}{\lim }\frac{\tan (kx)-kx}{x^3}$$

and state for which $k$ it exists.

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

$$\underset{x\to 0}{\lim }\frac{kx-\ln (1+x)}{x^2}$$

Determine

$$\underset{x\to 1}{\lim }\frac{x^k-1}{x-1}$$ and state for which $k$ this limit exists.

Find all $k$ such that the limit exists:

$$\underset{x\to 0}{\lim }\frac{x-\sin (kx)}{x^3}$$

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

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

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

$$\underset{x\to 0}{\lim }\frac{k^{2}x-kx}{4x^2}.$$

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

$$\underset{t\to \frac{\pi }{2}}{\lim }\frac{\cos (kt)}{t-\frac{\pi }{2}}$$

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

$$\underset{x\to 0}{\lim }\frac{k{x}^2+x\sin x-x^2}{x^3}$$

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

$$\underset{x\to 0}{\lim }\frac{(x-\sin x)-k{x}^3}{x^5}.$$

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

$$\underset{x\to 0}{\lim }\frac{a{x}^3+b{x}^2}{x-\sin x}.$$