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Example Questions
Example Question #401 : Finding Limits And One Sided Limits
Example Question #402 : Finding Limits And One Sided Limits
Example Question #401 : Finding Limits And One Sided Limits
Example Question #441 : Limits
Example Question #441 : Limits
Example Question #406 : Finding Limits And One Sided Limits
Example Question #407 : Finding Limits And One Sided Limits
Example Question #451 : Limits
Evaluate the 1-sided limits
1
0
If we were to look at a plot of the function, the limit would be obvious. Drawing the graph would be difficult without a graphing device. We can overcome this by first defining a new variable and assessing how it behaves as the original variable approaches from the right.
Let,
Therefore, as
We notice that the function in terms of has a graph that is much easier to draw from recollection since we readily know how the natural logarithm behaves. In the plot of the natural log, the y-axis is a horizontal asympote, and the function becomes infinite in the negative direction as
becomes arbitrarily close to 0 from the right. Hence
.
The graph above is more familiar. This is not the graph of the original function , which is in terms of
The plot above represents the function defined in terms of the variable we defined,
.
Example Question #401 : Finding Limits And One Sided Limits
One-Sided Limits
Find
As gets closer to 3 (but remains larger than 3), then
gets closer to 0 (but remains a small positive number).
The numerator, , gets closer to 6.
So, is an arbitrarily large positive number.
Thus, we conclude that the limit is .
Example Question #401 : Finding Limits And One Sided Limits
One-Sided Limits
Find
As gets closer to 3 (but remains smaller than 3), then
gets closer to 0 (but remains a small negative number).
The numerator, , gets closer to 6. So,
is an arbitrarily large negative number.
Thus, we conclude that the limit is .
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