You need to evaluate the limit $\lim_{{x \to 1}} e^{7x^2 - 7x}$ using continuity.

Solution:

We can solve this by substituting $x = 1$ directly into the expression, as the function is continuous. Since exponential functions are continuous everywhere, we can compute the value of the exponent at $x = 1$ directly.

1. Substitute $x = 1$ into the exponent: $7(1)^2 - 7(1) = 7 - 7 = 0$

2. Now the limit becomes: $e^0 = 1$

Thus, the value of the limit is:

$\lim_{{x \to 1}} e^{7x^2 - 7x} = 1$

Would you like more details on any of the steps? Here are some related questions to explore further:

1. What is the general process for solving limits using continuity?
2. How does the continuity of the exponential function help in limit evaluation?
3. Can a function still be continuous if it's defined piecewise?
4. How would the result change if the exponent had a discontinuity at $x = 1$?
5. What are the key differences between continuous and differentiable functions?

Tip: When dealing with limits, always check for continuity at the point you're approaching. If the function is continuous, you can substitute the value directly!