I need step-by-step help for solving the lim x -> 0 for the function sin(x)/(x + tan(x))

I can do simpler ones, but this one throws me off.

Thanks.

use L'Hopital's rule

lim sinx/(x + tanx) , as x---0
= lim cosx/( 1 + sec^2 x) as x --->0
= 1/(1+1)
= 1/2

test with your calculator:
enter .0000001 and store in memory of calculator
Make sure calculator is set to RAdians

evaluate:
RCL M
sin
=
÷
(RCL M
+
TAN
RCL M
)
=

you should get .5 or very close to .5

I use such a method for any limit question, to either check my answer or get a prediction of my answer

Perfect, thanks.

Sure, I can help you with solving the limit as x approaches 0 for the function sin(x)/(x + tan(x)). Here are the steps:

Step 1: Rewrite the expression.
We can rewrite the expression as sin(x)/x * 1/(1 + tan(x)/x).

Step 2: Identify the individual limits.
We need to find the limit as x approaches 0 for each term separately: sin(x)/x and 1/(1 + tan(x)/x).

Step 3: Evaluate the first term limit.
The limit of sin(x)/x as x approaches 0 is a well-known limit and equals 1. You can either apply L'Hôpital's Rule or use a trigonometric identity.

Step 4: Simplify the second term.
Let's focus on the term 1/(1 + tan(x)/x). As x approaches 0, tan(x)/x approaches 0 as well. So, we can simplify this term to 1/(1 + 0), which is equal to 1.

Step 5: Combine the individual limits.
Now that we have found the limits of both individual terms, we can multiply them together. 1 * 1 equals 1.

Step 6: Final Result.
The limit as x approaches 0 for the function sin(x)/(x + tan(x)) is 1.

In summary, to solve the limit as x approaches 0 for the function sin(x)/(x + tan(x)), we rewrote the expression, evaluated the individual limits, simplified the terms, and then combined the results.