Question

How do you find the limit of `(1-e^(-2x))/(sec(x))` as x approaches 0?

Answer 1

`0`

Explanation:

Step 1:
`(1-e^(-2x))/sec(x) rarr (1-e^(-2x))cos(x)`

because `sec(x)=1/cos(x)`

Step 2:
the cosine function is known to have a range of `-1<=cos(x)<=1`

So we can write:
`-1(1-e^(-2x))<=(1-e^(-2x))cos(x)<=1(1-e^(-2x))`

Step 3:
`lim_(xrarr0)-(1-e^(-2x)) rarr lim_(xrarr0)-1+1/e^(2x)= -1+1/e^(0)=-1+1/1=0`

Step 4:
`lim_(xrarr0)(1-e^(-2x)) rarr lim_(xrarr0)1-1/e^(2x)= 1-1/e^(0)=1-1/1=0`

Hence, `lim_(xrarr0)(1-e^(-2x))cos(x)=color(blue)0`

Recall:
According to the color(black)("Sandwich theorem"), if `a(x)<=f(x)<=b(x)`
And `lim_(x rarr c)a(x)=K=lim_(x rarr c)b(x)`
this implies that `lim_(xrarrc)f(x)= K`