Question

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

`f(x) = {{:(x",", x ≤ 1),(x^2",", x > 1):}`

Answer 1

To find the left limit  of `f(x)` at x = 1

Put x = 1 – h

When x → 1

We have h

Where x = 1 – h

We have x  > 1

∴ `f(x)` = x¹

`f"'"(1^-) =  lim_("h" - 0) (f(1 - "h") - (1))/(1 - "h" - 1)`

= `lim_("h" -> 0) (1 - "h" - 1)/(- "h")`

= `lim_("h" > 0) (- "h")/(- h")` = 1  ........(1)

To find the right limit  of `f(x)` at x = 1

Put x = 1 + h

When x → 1

We have h → 0

Where x = 1 + h

We have x > 1

∴ `f(x)` = x² 

`f"'"(1^+) lim_("h" -> 0) (f(1 + "h") - f(1))/(1 + "h" - 1)`

= `lim_("h" - 0) ((1 + "h")2 - 1^2)/"h"`

= `lim_("h" - 0) (1 + 2"h" + "h"^2 - 1)/"h"`

= `lim_("h" -> 0) (2"h" + "h"^2)/"h"`

= `lim_("h" -> 0) ("h"(2 + "h"))/"h"`

= `lim_("h" -> 0) (2 + "h")`

`f"'"(1^+) = 2 + 0` = 2   ..........(2)

From equation (1) and (2) we get

`f"'"(1^+)  ≠  f"'"(1^+)`

∴ f(x) is not differentiable at x = 1.