+130511 Note that we are taking the limit as x approaches 0 from the right.......
Using the Puiseux series expansion for the first six terms of sin (sqrt(x)), we have :
lim x → 0 [sqrt(x)-x^(3/2)/6+x^(5/2)/120-x^(7/2)/5040+x^(9/2)/362880-x^(11/2)/39916800] / x
Dividing each term by x and taking the limit as x → 0 on the last five terms will result in 0
Dividing the first term by x , we have
lim x → 0 1 / sqrt(x)
As x → 0, the denominator gets extremely "small" in comparison to the numerator .... so the function → +infinity as x → 0 ....i.e., .......the limit does not exist
Here's the graph that confirms this ..... https://www.desmos.com/calculator/tzfgdcaps7
+130511 Note that we are taking the limit as x approaches 0 from the right.......
Using the Puiseux series expansion for the first six terms of sin (sqrt(x)), we have :
lim x → 0 [sqrt(x)-x^(3/2)/6+x^(5/2)/120-x^(7/2)/5040+x^(9/2)/362880-x^(11/2)/39916800] / x
Dividing each term by x and taking the limit as x → 0 on the last five terms will result in 0
Dividing the first term by x , we have
lim x → 0 1 / sqrt(x)
As x → 0, the denominator gets extremely "small" in comparison to the numerator .... so the function → +infinity as x → 0 ....i.e., .......the limit does not exist
Here's the graph that confirms this ..... https://www.desmos.com/calculator/tzfgdcaps7
