Evaluate the limit.
First, we multiply and divide by the conjugate of the expression, then simplify and take the limit,
Evaluate the limit.
First, we multiply and divide by the conjugate of the expression, then simplify and take the limit,
Evaluate the limit.
First, we make the substitution . Then
as
so we have
Evaluate the limit.
First,
Then, we multiply inside the limit by since
,
Evaluate the limit.
We recall the definition of the hyperbolic cosine in terms of the exponential,
Using this we compute,
Evaluate the limit.
Both the numerator and denominator go to 0 as so we apply L’Hopital’s rule,
Evaluate the limit.
We use the trig identity and then use L’Hopital’s rule to evaluate,
Evaluate the limit.
Let , then
as
and we have
Evaluate the limit.
First, we pull an out of
and an
out of
to write
Then we have
Evaluate the limit.
We write and apply L’Hopital’s rule to solve
Evaluate the limit.
Let , then
as
. Making this substitution and using L’Hopital’s rule we have