Suppose lim x->0 {g(x)-g(0)} / x = 1.
It follows necesarily that
a. g is not defined at x=0
b. the limit of g(x) as x approaches equals 1
c.g is not continuous at x=0
d.g'(0) = 1
The answer is d, can someone please explain how?
Thanks.
lim x->0 {g(x)-g(0)} / x = 1.
You can use the definition of the derivative:
g'(x) = Lim h--> [g(x+h) - g(x)]/h
Take x = 0:
g'(0) = Lim h--> [g(h) - g(0)]/h
And h is just a "dummy variable" whose name doesn't matter :)
Now, substitute the given limit:
g'(0) = Lim h--> [g(h) - g(0)]/h
= Lim h--> [1*h - 0]/h
= Lim h--> 1
= 1
Therefore, g'(0) = 1.
In the given equation, we have:
lim x->0 {g(x)-g(0)} / x = 1.
To find out which statement necessarily follows, let's examine each option:
a. g is not defined at x=0: This statement cannot be concluded from the given equation. The equation only provides information about the limit of the expression, not the definition of g(x) at x=0.
b. The limit of g(x) as x approaches 0 equals 1: This statement also cannot be concluded from the given equation. The equation only gives information about the difference quotient and its limit, but it does not provide information about the limit of g(x) itself.
c. g is not continuous at x=0: This statement also cannot be concluded from the given equation. The limit of the difference quotient being 1 does not necessarily imply that g(x) is not continuous at x=0.
d. g'(0) = 1: This statement can be concluded from the given equation. When we apply the definition of the derivative:
g'(0) = Lim h--> [g(h) - g(0)]/h.
Substituting x=0, we have:
g'(0) = Lim h--> [g(0+h) - g(0)]/h.
Since the limit of the difference quotient equals 1, we can conclude that g'(0) = 1.
Therefore, the correct answer is d. g'(0) = 1.
To prove that the answer is d, we need to show that g'(0) = 1.
The given expression can be rewritten as:
lim x->0 {g(x) - g(0)} / x = 1
We can rewrite the numerator of the expression in terms of a difference quotient:
lim x->0 {g(x) - g(0)} = x * 1
Now, let's apply the definition of the derivative as mentioned in the question:
g'(0) = lim h->0 {g(0 + h) - g(0)} / h
Notice that we have essentially substituted x with 0 and replaced h in the denominator with x. This is to align our expression with the definition of the derivative.
Applying the definition, we can simplify the expression as follows:
g'(0) = lim h->0 {g(h) - g(0)} / h
Comparing this expression with the given expression, we see that they are the same, except for the limit variable being x or h. But since h is just a dummy variable, its name doesn't matter.
From this comparison, we can conclude that g'(0) = 1, which matches option d.
Therefore, the correct answer is d.