Question

Select the correct answer from the given alternatives.

`lim_(x -> 3) [(5^(x - 3) - 4^(x - 3))/(sin(x - 3))]` =

Options

• log 5 – 4

• `log  5/4`

• `log5/log4`

• `log5/4`

Answer 1

`log  5/4`

Explanation;

`lim_(x -> 3) (5^(x - 3) - 4^(x - 3))/(sin(x - 3))`

Put x – 3 = h

∴ x = 3 + h

As → 3, h → 0

∴ Required limit

= `lim_("h" -> 0) (5^"h" - 4^"h")/(sin "h")`

= `lim_("h" -> 0) ((5^"h" - 1) - (4^"h" - 1))/sin"h"`

= `lim_("h" -> 0) (((5^"h" - 1))/"h" - ((4^"h" - 1))/"h")/(lim_("h" -> 0) sin"h"/"h"`

= `(log5 - log4)/1  ...[lim_(x -> 0) ("a"^x - 1)/x = log"a"]`

= `log(5/4)`