Estimate the following limit using graphs or tables. In (1 + h) lim h h?0 In (1 + h) lim h h- (Type an integer or decimal rounded to the nearest hundredth as needed.)
The Correct Answer and Explanation is:
To estimate the limit limh→0ln(1+h)h,\lim_{h \to 0} \frac{\ln(1 + h)}{h},
we can use graphs or tables. But first, let’s understand what this limit represents and how we can estimate it.
Correct Answer:
1.00\boxed{1.00}
Explanation
This limit is a well-known fundamental limit in calculus: limh→0ln(1+h)h=1.\lim_{h \to 0} \frac{\ln(1 + h)}{h} = 1.
We can understand and estimate this using a table of values or a graph of the function f(h)=ln(1+h)h.f(h) = \frac{\ln(1 + h)}{h}.
Let’s look at some values of hh close to 0:
| hh | ln(1+h)h\frac{\ln(1 + h)}{h} |
|---|---|
| -0.1 | 0.9486 |
| -0.01 | 0.9950 |
| -0.001 | 0.9995 |
| 0.001 | 1.0005 |
| 0.01 | 1.0050 |
| 0.1 | 1.0486 |
As hh gets closer to 0 from both sides (positive and negative), the values of the function approach 1. This strongly suggests that: limh→0ln(1+h)h=1.\lim_{h \to 0} \frac{\ln(1 + h)}{h} = 1.
Why Is This Important?
This limit is the derivative of ln(x)\ln(x) at x=1x = 1. In calculus, the derivative is defined as: f′(a)=limh→0f(a+h)−f(a)h.f'(a) = \lim_{h \to 0} \frac{f(a + h) – f(a)}{h}.
For f(x)=ln(x)f(x) = \ln(x), we get: ln′(1)=limh→0ln(1+h)−ln(1)h=limh→0ln(1+h)h.\ln'(1) = \lim_{h \to 0} \frac{\ln(1 + h) – \ln(1)}{h} = \lim_{h \to 0} \frac{\ln(1 + h)}{h}.
So, the value of this limit tells us the rate of change of the natural logarithm at 1, and it is exactly 1. This is a key concept in understanding how logarithmic functions behave near 1.
Final Answer:
1.00\boxed{1.00}
