Estimate the limit numerically.
lim as x→(approaches to)5 6e^(x − 5)
How do I solve it? I don't plug in the numbers do I?
Then how do I do this as well? What should I do to solve
lim x→(approaches to)− infinity 6xe^(7x)
Do I disregard the other numbers after 6x?
Thank you
Go on :
wolframalpha dot com
When page be open in rectangle type :
limit 6e^(x − 5) as x->5
and click option =
When you see result click option:
Show steps
For your first limit, as x approached 5 e^x will equal 1 and 6*1 is 6.
For your second one is infinity * infinity= infinity
And yes, the steps are on wolframalpha, but try to figure these out yourself first
To estimate the limit numerically, you can use a calculator or a software tool that can handle mathematical functions. Here's how you can solve each of the given limits:
1. lim as x → 5 of 6e^(x − 5):
To evaluate this limit, you can substitute values of x that approach 5 into the function and observe the corresponding output values. Start by choosing values slightly less than 5 (e.g., 4.9, 4.99, 4.999) and values slightly greater than 5 (e.g., 5.1, 5.01, 5.001). Calculate the function value for each of these x-values and observe the trend. As x gets closer to 5 from both sides, the function values will approach a certain value. This value is the numerical estimation of the limit.
For example, let's calculate the function values for a few x-values:
- For x = 4.9: f(4.9) = 6e^(4.9 - 5) = 5.7725
- For x = 4.99: f(4.99) = 6e^(4.99 - 5) = 5.9344
- For x = 4.999: f(4.999) = 6e^(4.999 - 5) = 5.9966
- For x = 5.1: f(5.1) = 6e^(5.1 - 5) = 6.2801
- For x = 5.01: f(5.01) = 6e^(5.01 - 5) = 6.1203
- For x = 5.001: f(5.001) = 6e^(5.001 - 5) = 6.0079
As x approaches 5, the function values seem to be approaching a value around 6. So, numerically, the estimated limit is approximately 6.
2. lim as x → -∞ of 6xe^(7x):
In this case, you cannot substitute values of x directly, as it approaches negative infinity. Instead, you need to consider the behavior of the function as x becomes extremely negative.
Notice that in the given expression, the term '6x' dominates over 'e^(7x)' as x approaches negative infinity. This is because the exponential function decreases rapidly as x goes to negative infinity, making it negligible compared to the linear term.
Therefore, you can simplify the function to approximate the limit: lim as x → -∞ of 6xe^(7x) ≈ lim as x → -∞ of 6x.
Now, consider the behavior of '6x' as x approaches negative infinity. Since x is approaching negative infinity, '6x' will be negative and its absolute value will grow without bound. Therefore, the limit of '6x' as x approaches negative infinity is negative infinity.
Hence, numerically, the estimated limit is -∞ (negative infinity).
Note: Disregarding other numbers after '6x' is appropriate because they become negligible compared to the dominant term as x approaches negative infinity.