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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!

Use continuity to evaluate the limit: lim (x→1) e^(7x^2 - 7x).

Learn how to evaluate the limit lim (x→1) e^(7x^2 - 7x) using the concept of continuity. This problem involves understanding exponential functions and applying the continuity theorem to find the limit by direct substitution. Suitable for high school calculus students or introductory college-level calculus.

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