Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine its continuity.

Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine its continuity.

Step 1: Understand the limit we want to evaluate: lim (x -> 0) (sin(5x)/x).
Step 2: Recognize that we can use a limit property: lim (u -> 0) (sin(u)/u) = 1.
Step 3: In our case, let u = 5x. As x approaches 0, u also approaches 0.
Step 4: Rewrite the limit in terms of u: lim (x -> 0) (sin(5x)/x) = lim (u -> 0) (sin(u)/(u/5)).
Step 5: Simplify the expression: lim (u -> 0) (sin(u)/(u/5)) = lim (u -> 0) (5 * sin(u)/u).
Step 6: Apply the limit property: lim (u -> 0) (sin(u)/u) = 1, so we have 5 * 1 = 5.
Step 7: Conclude that lim (x -> 0) (sin(5x)/x) = 5.
Step 8: To check continuity, note that the limit exists and is finite, and the function sin(5x)/x is defined at x = 0 (we can define it as 5).
Step 9: Therefore, the function is continuous at x = 0.

Limit Evaluation – Understanding how to evaluate limits, particularly using known limit properties such as lim (x -> 0) (sin(kx)/x) = k.
Continuity – Determining the continuity of a function at a point, specifically checking if the limit equals the function's value at that point.