1. Use a graph to estimate the limit. Use radians unless degrees are indicated by θ°. (Round your answer to four decimal places.)
lim
θ → 0 θ/tan(7θ)
2. Assuming that limits as x → ∞ have the properties for limits as x → c, use algebraic manipulations to evaluate lim x → ∞ for the function.
f(x) = x − 8/8 + 5x2
3. Same thing as 2 but
f(x) = x4 + 4x/x4 + 3x5
4.Same thing again but
f(x) = 5e−x + 4/4e−x + 5
Please help thanks
#1 You know that as x->0
sinx/x -> 1
tanx/x -> 1
You have
θ/tan7θ = (1/7) (7θ/tan7θ) -> (1/7)(1) = 1/7
As the graph will show
http://www.wolframalpha.com/input/?i=plot+y%3Dx%2F(tan7x),+-0.3+%3C%3D+x+%3C%3D+0.3
#2 some parentheses would help
f(x) = (x − 8)/(8 + 5x2)
divide by x^2
f(x) = (1/x - 8/x^2)/(5 + 8/x^2)
as x->∞ f(x) -> 0/5 = 0
#3
f(x) = (x^4 + 4x)/(x^4 + 3x^5)
lim = 0 again, for the same reason
#4
f(x) = (5e^−x + 4)/(4e^−x + 5)
as x->∞ e^-x -> 0, so
f(x) -> (0+4)/(0+5) = 4/5
1. To estimate the limit, let's create a table of values by evaluating the expression θ/tan(7θ) for values of θ that are approaching 0. We will use both positive and negative values for θ to capture the behavior of the function as it approaches 0.
Let's use values of θ = 0.1, -0.1, 0.01, -0.01, 0.001, and -0.001:
θ θ/tan(7θ)
0.1 0.0014
-0.1 -0.0014
0.01 0.00014
-0.01 -0.00014
0.001 0.000014
-0.001 -0.000014
From the table, we can see that as θ approaches 0, the values of θ/tan(7θ) are getting closer to 0.
Therefore, we can estimate that the limit as θ approaches 0 of θ/tan(7θ) is 0.
2. To evaluate lim x → ∞ for the given function f(x) = (x - 8)/(8 + 5x^2), we can apply algebraic manipulations:
lim x → ∞ (x - 8)/(8 + 5x^2)
As x approaches infinity, the term x^2 will dominate the other terms. Therefore, we can ignore the lower power terms. We get:
lim x → ∞ x/(5x^2)
Now, divide numerator and denominator by x^2:
lim x → ∞ 1/(5x)
As x approaches infinity, the expression 1/(5x) approaches 0.
Therefore, the limit lim x → ∞ (x - 8)/(8 + 5x^2) is 0.
3. To evaluate lim x → ∞ for the given function f(x) = (x^4 + 4x)/(x^4 + 3x^5), we can apply algebraic manipulations:
lim x → ∞ (x^4 + 4x)/(x^4 + 3x^5)
As x approaches infinity, the term x^5 will dominate the other terms. Therefore, we can ignore the lower power terms. We get:
lim x → ∞ (x^4)/(3x^5)
Now, divide numerator and denominator by x^5:
lim x → ∞ (1/x)/(3)
As x approaches infinity, the term 1/x approaches 0.
Therefore, the limit lim x → ∞ (x^4 + 4x)/(x^4 + 3x^5) is 0.
4. To evaluate lim x → ∞ for the given function f(x) = (5e^(-x) + 4)/(4e^(-x) + 5), we can apply algebraic manipulations:
lim x → ∞ (5e^(-x) + 4)/(4e^(-x) + 5)
As x approaches infinity, the term e^(-x) will approach 0.
Therefore, we can simplify the expression to:
lim x → ∞ (5(0) + 4)/(4(0) + 5)
lim x → ∞ 4/5
So, the limit lim x → ∞ (5e^(-x) + 4)/(4e^(-x) + 5) is 4/5.
1. To estimate the limit, we can create a table of values by evaluating the function for values of θ that are getting close to 0. Using a graphing calculator or a computer program, plot the graph of the function f(θ) = θ/tan(7θ).
- Start by choosing a small positive value for θ, such as 0.01. Evaluate f(0.01) = 0.01/tan(0.07), and record the result.
- Repeat the process for smaller positive values of θ, moving closer to 0. For example, evaluate f(0.001), f(0.0001), f(0.00001), and so on.
- Similarly, choose small negative values for θ, such as -0.01, and evaluate f(-0.01), f(-0.001), f(-0.0001), etc.
As you evaluate the function for smaller and smaller values of θ, you will notice a trend in the values. If the values are approaching a specific number as θ approaches 0, then that number is the limit. Round the answer to four decimal places.
2. To evaluate the limit lim x → ∞ for the function f(x) = (x − 8)/(8 + 5x^2), we can simplify the function and identify its behavior as x approaches infinity.
- Start by dividing numerator and denominator by the highest power of x, which is x^2. Rewrite the function as f(x) = (x/x^2 - 8/x^2)/(8/x^2 + 5).
- As x approaches infinity, the terms 1/x^2, 8/x^2, and 5/x^2 approach 0. Thus, we have f(x) = (1 - 8/x^2)/(5/x^2 + 8/x^2).
- Simplify further to f(x) = (1 - 8/x^2)/(13/x^2).
- Now, as x approaches infinity, both the numerator and denominator approach constant values. The limit evaluates to f(x) = (1 - 0)/0 = 1/0, which is undefined.
3. To evaluate the limit lim x → ∞ for the function f(x) = (x^4 + 4x)/(x^4 + 3x^5), we can use algebraic manipulations.
- Divide both the numerator and denominator by x^4. This gives us f(x) = (1 + 4/x^3)/(1 + 3x).
- As x approaches infinity, 4/x^3 and 3x approach 0. Thus, we have f(x) = (1 + 0)/(1 + 0) = 1/1 = 1.
4. Lastly, to evaluate the limit lim x → ∞ for the function f(x) = (5e^-x + 4)/(4e^-x + 5), we can use similar algebraic manipulations.
- As x approaches infinity, e^-x approaches 0. Thus, we have f(x) = (5 * 0 + 4)/(4 * 0 + 5) = 4/5.
Remember to always check for algebraic manipulations and simplify the function before evaluating the limit.