Evaluate the limits without using a calculator. Justify all steps in the solutions.
lim x-->0 x^7cos(3/x)
To evaluate the limit lim(x→0) x^7cos(3/x) without using a calculator, we can use the properties of limits and some trigonometric identities. Let's break down the process step by step:
Step 1: Determine the form of the limit
The given limit has a product of two functions, x^7 and cos(3/x). To evaluate this limit, we need to evaluate the limits of these individual factors.
Step 2: Evaluate the limit of x^7
As x approaches 0, x^7 approaches 0^7, which is simply 0.
Step 3: Evaluate the limit of cos(3/x)
To evaluate the limit as x approaches 0 of cos(3/x), we use the fact that cos(θ) is a bounded function, meaning its values lie between -1 and 1 for any angle θ. Since the argument of the cosine function is tending to infinity (3/x approaches infinity as x approaches 0), we can say that cos(3/x) is bounded between -1 and 1.
Step 4: Multiply the limits
We have determined that the limit of x^7 is 0, and the limit of cos(3/x) is bounded between -1 and 1. Thus, when we multiply these two limits, we obtain:
lim(x→0) x^7cos(3/x) = 0 * (a bounded value between -1 and 1) = 0.
Step 5: Justify the steps
In this solution, we justified each step as follows:
- In Step 2, we used the concept that any non-zero number raised to the power of 0 is equal to 1.
- In Step 3, we utilized the fact that the cosine function is bounded between -1 and 1 for any angle.
- In Step 4, we employed the algebraic property that 0 multiplied by any number is equal to 0.
Therefore, the limit of lim(x→0) x^7cos(3/x) is 0.