QuestionFind the limit: $\lim_{x \rightarrow 0} f(x)$ where $f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)= x \cos \left(\frac{1}{x^{}}\right)$ . We are looking for the limit as $x$ approaches $0$ .

STEP 2

The limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value. In this case, we are looking for the limit as $x$ approaches $0$ .
$\lim{{x \to0}} f(x) = \lim{{x \to0}}2x \cos \left(\frac{1}{x^{2}}\right)$

STEP 3

The limit of a product is the product of the limits, so we can split this limit into two parts.
$\lim{{x \to0}}2x \cos \left(\frac{1}{x^{2}}\right) = \lim{{x \to0}}2x \times \lim{{x \to0}} \cos \left(\frac{1}{x^{2}}\right)$

STEP 4

The limit of $2x$ as $x$ approaches $0$ is simply $0$ . $\lim{{x \to0}}2x =0$

STEP 5

The limit of $\cos \left(\frac{1}{x^{2}}\right)$ as $x$ approaches $0$ is not defined because the cosine function oscillates between $-1$ and $1$ as $x$ approaches $0$ . However, this does not affect the overall limit because it is being multiplied by $0$ .

STEP 6

Multiply the two limits together to find the overall limit.
$\lim{{x \to0}} f(x) =0 \times \text{undefined} =0$ So, the limit of the function $f(x)=2 x \cos \left(\frac{1}{x^{2}}\right)$ as $x$ approaches $0$ is $0$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms