Answer:
[tex] \displaystyle \sec ^{2} (a) [/tex]
Step-by-step explanation:
we are given a limit
[tex] \displaystyle \lim_{x \to a} \frac{ \tan(x) - \tan(a) }{ \tan(x - a) } [/tex]
and said to compute without L'Hopitâl rule
if we substitute a for x directly we'd get
[tex] \displaystyle \frac{ \tan(a) - \tan(a) }{ \tan(a - a) } [/tex]
[tex] \displaystyle \: \frac{0}{0} [/tex]
which is indeterminate
so we have to do it differently
recall trigonometric indentity
[tex]\sf \displaystyle \tan(A\pm B)=\dfrac{\tan(A)\pm \tan(B)}{1\mp \tan(A)\tan(B)}[/tex]
using the identity we get
[tex] \displaystyle \lim_{x \to a} \frac{ \tan(x) - \tan(a) }{ \dfrac{\tan(x) - \tan(a)}{1 + \tan(x)\tan(a)}} [/tex]
simplify complex fraction:
[tex] \sf \displaystyle \lim_{x \to a} \cancel{\tan(x) - \tan(a)} \times \frac{1 + \tan(x) \tan(a) }{ \cancel{\tan(x) - \tan(a) } }[/tex]
[tex] \displaystyle \lim_{x \to a} 1 + \tan(x) \tan(a) [/tex]
now we can substitute a for x:
[tex] \displaystyle 1 + \tan(a) \tan(a) [/tex]
simplify multiplication:
[tex] \displaystyle 1 + \tan ^{2} (a)[/tex]
recall trigonometric indentity:
[tex] \displaystyle \sec ^{2} (a) [/tex]
